The elements of non-Euclidean geometry · 2009. 11. 6. · theelementsof non-euclideangeometry...

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THE ELEMENTS OF

NON-EUCLIDEAN GEOMETRY

JULIAN LOWELL COOLIDGE Ph.D.ASSISTANT PBOFESSOR OF MATBEHATICS

IV HABVAVD UNIVCK8ITV

OXFORD

AT THE CLARENDON PRESS

1909

HENRY FROWDE, M.A.

PUBliISHEB TO THE UNIVEBSITT OF OXFOBB

LONDON, BDINBURGH, NBW TORE

TORONTO AND HBLBOURNE

PREFACE

The heroic age of non-euclidean geometry is passed.

It is long since the days when Lobatchewsky timidly

referred to his system as an 'imaginary geometry',

and the new subject appeared as a dangerous lapse

from the orthodox doctrine of Euchd. The attempt to

prove the parallel axiom by means of the other usual

assumptions is now seldom undertaken, and those whodo undertake it, are considered in the class with

circle-squarers and searchers for perpetual motion—sad

by-products of the creative activity of modern science.

In this, as in all other changes, there is subject both

for rejoicing and regret. It is a satisfaction to a writer

on non-euclidean geometry that he may proceed at

once to his subject, without feeling any need to justify

himself, or, at least, any more need than any other

who adds to our supply of books. On the other hand,

he will miss the stimulus that comes to one who feels

that he is bringing out something entirely new and

strange. The subject of non-eucUdean geometry is, to

the mathematician, quite as well established as any

other branch of mathematical science ; and, in fact, it

may lay claim to a decidedly more solid basis than

some branches, such as the theory of assemblages, or

the analysis situs.

Kecent books dealing with non-euchdean geometry

fall naturally into two classes. In the one we find

the works of Killing, Liebmann, and Manning,* who* Detailed references given later.

a2

4 PREFACE

wish to build up certain clearly conceived geometrical

systems, and are careless of the details of the founda-

tions on which all is to rest. In the other category

are Hilbert, Vahlen, Veronese, and the authors of

a goodly number of articles on the foundations of

geometry. These writers deal at length mth the

consistency, significance, and logical independence of

their assimiptions, but do not go very far towards

raising a superstructure on any one of the foundations

suggested.

The present work is, in a measure, an attempt to

unite the two tendencies. The author's own interest,

be it stated at the outset, lies mainly in the fruits,

rather than in the roots ; but the day is past when the

matter of axioms may be dismissed with the remark

that we ' make all of Euclid's assumptions except the

one about parallels'. A subject like ours must be

built up from explicitly stated assumptions, and nothing

else. The author would have preferred, in the first

chapters, to start from some system of axioms already

published, had he been fanuliar with any that seemed to

him suitable to establish simultaneously the euclidean

and the principal non-eucUdean systems in the way that

he wished. The system ofaxioms here used is decidedly

more cumbersome than some others, but leads to the

desired goal.

There are three natural approaches to non-euclidean

geometry. (1) The elementary geometry of point, line,

and distance. This method is developed in the open-

ing chapters and is the most obvious. (2) Projective

geometry, and the theory of transformation groups.

This method is not taken up until Chapter XVIII, not

because it is one whit less important than the first, but

PREFACE 5

"because it seemed better not to interrupt the natural

course of the narrative by interpolating an alternative

beginning. (3) Differential geometry, with the con-

cepts of distance^lement, extremal, and space constant.

This method is explained in the last chapter, XIX.The author has imposed upon himself one or two

very definite limitations. To begin with, he has not

gone beyond three dimensions. This is because of his

feeling that, at any rate in a first study of the subject, the

gain in generality obtained by studying the geometry

of ^-dimensions is more than offset by the loss of

clearness and naturalness. Secondly, he has confined

himself, almost exclusively, to what may be called the

* classical ' non-euclidean systems. These are muchmore closely aUied to the euclidean system than are

any others, and have by far the most historical impor-

tance. It is also evident that a system which gives

a simple and clear interpretation of ternary and qua-

ternary orthogonal substitutions, has a totally different

sort of mathematical significance from, let us say, one

whose points are determined by numerical values in

a non-archimedian number system. Or again, a non-

euchdean plane which may be interpreted as a surface

of constant total curvature, has a more lasting geo-

metrical importance than a non-desarguian plane that

cannot form part of a three-dimensional space.

The majority of material in the present work is,

naturally, old. A reader, new to the subject, may find

it wiser at the first reading to omit Chapters X, XV,

XVI, XVIII, and XIX. On the other hand, a reader

already somewhat familiar with non-euclidean geo-

metry, may find his greatest interest in Chapters Xand XVI, which contain the substance of a number of

6 PREFACE

recent papers on the extraordinary line geometry of

non-euclidean space. Mention may also be madeof Chapter XTV which contains a number of neat

formulae relative to areas and volumes published

many years ago by Professor d'Ovidio, which are not,

perhaps, very familiar to English-speaking readers,

and Chapter XIII, where Staude's string construction

of the ellipsoid is extended to non-eucHdean space.

It is hoped that the introduction to non-euchdean

differential geometry in Chapter XV may prove to

be more comprehensive than that of Darboux, andmore comprehensible than that of Bianchi.

The author takes this opportunity to thank his

colleague, Assistant-Professor Whittemore, who hasread in manuscript Chapters XV and XIX. He wouldalso oflter afiPectionate thanks to his former teachers,

Professor Eduard Study of Bonn and Professor CorradoSegre of Turin, and all others who have aided andencouraged (or shall we say abetted?) him in the

present work.

TABLE OF CONTENTS

CHAPTER I

FOUNDATION FOR METRICAL GEOMETRY IN A LIMITED REGIONPAGE

Fundamental assumptions and definitions 13

Sums and differences of distances 14

Serial arrangement of points on a line 15

Simple descriptive properties of plane and space . .17

CHAPTER nCONGRUENT TRANSFORMATIONS

Axiom of continuity 23

Division of distances . 23

Measure of distance 26

Axiom of congruent transformations 29

Definition of angles, their properties 30

' Comparison of triangles 32

Side of a triangle not greater than sum of other two ... 35

Comparison and measurement of angles 37

Nature of the congruent group 38

Definition of dihedral angles, their properties 39

CHAPTER mTHE THREE HYPOTHESES

A variable angle is a continuous function of a variable distance . 40

Saccheri's theorem for isosceles birectangular quadrilaterals . . 43

The existence of one rectangle implies the existence of an infinite

number 44

Three assumptions as to the sum of the angles of a right triangle . 45

Three assumptions as to the sum of the angles of any triangle, their

categorical nature 46

Definition of the euclidean, hyperbolic, and elliptic hypotheses 46

Geometry in the infinitesimal domain obeys the euclidean hypothesis 47

CHAPTER IV

TRIGONOMETRIC FORMULAELimit of ratio of opposite sides of diminishing isosceles quadri-

lateral 48

Continuity of the resulting function 50

Its functional equation and solution 51

Functional equation for the cosine of an angle ... 54

Non-euclidean form for the pythagorean theorem . . . 55

Trigonometric formulae for right and oblique triangles . . 57

8 CONTENTS

CHAPTER VANALYTIC FORMULAE

FAOEDirected distances ^'^

Group of translations of a line 62

Positive and negative directed distances 64

Coordinates of a point on a line 64

Coordinates of a point in a plane 65

Finite and infinitesimal distance formulae, the non-euclidean plane

as a surface of constant Gaussian curvature .... 65

Equation connecting direction cosines of a line . . 67

Coordinates of a point in space 68

Congruent transformations and orthogonal substitutions . 69

Fundamental formulae for distance and angle 70

CHAPTER VI

CONSISTENCE AND SIGNIFICANCE OF THE AXIOMS

Examples of geometries satisfying the assumptions made 72

Relative independence of the axioms 74

CHAPTER VnGEOMETRIC AND ANALYTIC EXTENSION OF SPACE

PossibUity of extending a segment by a definite amount in theeuclidean and hyperbolic cases 77

Euclidean and hyperbolic space 77

Contradiction arising under the elliptic hypothesis.... 78

New assumptions identical with the old for limited region, but per-

mitting the extension of every segment by a definite amount . IS

Last axiom, free mobility of the whole system 80

One to one correspondence of point and coordinate set in euclideanand hyperbolic cases 81

Ambiguity in the eUiptic case giving rise to elliptic and spherical

geometry 81

Ideal elements, extension of all spaces to be real continua . . 84

Imaginary elements geometrically defined, extension of all spacesto be perfect continua in the complex domain .... 85

Cayleyan Absolute, new form for the definition of distance 88

Extension of the distance concept to the complex domain . . 89

Case where a straight line gives a maximum distance ... 91

CHAPTER VmGROUPS OF CONGRUENT TRANSFORMATIONS

Congruent transformations of the straight line . . . .94„ „ „ hyperbolic plane .... 94

CONTENTS 9

PAGECongruent transformations of the elliptic plane

„ „ „ enclidean plane .

„ „ „ hyperbolic space .

„ „ „ elliptic and spherical space

Clifford parallels, or paratactic lines ....The groups of right and left translations....Congruent transformations of euclidean space .

96

97

98

99

99

100

CHAPTER IXPOINT, LINE, AND PLANE, TREATED ANALYTICALLY

Notable points of a triangle in the non-euclidean plane . .101Analoga of the theorems of Menelaus and Ceva ... 104

Formulae of the parallel angle 106

Equations of parallels to a given line 107

Notable points ofa tetrahedron, and resulting desmic configurations 108

Invariant formulae for distance and angle of skew lines in line

coordinates" 110

Criteria for parallelism and parataxy in line coordinates. . .11.3

Relative moment of two directed lines 114

CHAPTER XHIGHER LINE-GEOMETRY

Linear complex in hyperbolic space 116

The cross, its coordinates 117

The use of the cross manifold to interpret the geometry of thecomplex plane 1 18

Chain, and chain surface 119

Hamilton's theorem 120

Chain congruence, synectic and non-synectic congruences 121

Dual coordinates of a cross in elliptic case .... 124

Condition for parataxy ... 125

Clifford angles 126

Chain and strip.... .... 128

Chain congruence 129

CHAPTER XITHE CIRCLE AND THE SPHERE

Simplest form for the equation of a circle . . . 131

Dual nature of the curve 131

Curvature of a circle 133

Radical axes, and centres of similitude 134

Circles through two points, or tangent to two lines . 135

Spheres 138

Poincare's sphere to sphere transformation from euclidean to non-

eucUdean space 189

10 CONTENTS

CHAPTER XII

CONIC SECTIONSPAOE

Classification of conies 142

Equations of central conic and Absolute 143

Centres, axes, foci, focal lines, directrices, and director points . 143

Relations connecting distances of a point from foci, directrices, &c.,

and their duals 144

Conjugate and mutually perpendicular lines through a centre . 148

Auxiliary circles .150Normals 152

Confocal and homothetic conies 152

Elliptic coordinates . . 152

CHAPTER Xni

QUADRIC SURFACES

Classification of quadrics 154

Central quadrics 157

Planes of circular section and parabolic section . .158Conjugate and mutually perpendicular lines through a centre 159

Confocal and homothetic quadrics 160

Elliptic coordinates, various forms of the distance element .161String construction for the ellipsoid 166

CHAPTER XIV

AREAS AND VOLUMES

Amplitude of a triangle 170

Relation to other parts 171

Limiting form when the triangle is infinitesimal .... 174

Deficiency and area 175

Area found by integration 176

Area of circle . 178

Area of whole elliptic or spherical plane 178

Amplitude of a tetrahedron .178Relation to other parts 179

Simple form for the differential of volume of a tetrahedron . . 181

Reduction to a single quadrature of the problem of finding thevolume of a tetrahedron 184

Volume of a cone of revolution 185

Volume of a sphere 186

Volume of the whole of elliptic or of spherical space . 186

CONTENTS 11

CHAPTER XV

INTRODUCTION TO DIFFERENTIAL GEOMETRY

PAGECurvature of a space or plane curve 187

Analoga of direction cosines of tangent, principal normal, andbinormal 189

Frenet's formulae for the non-euclidean case 190'

Sign of the torsion 191

Evolutes of a space curve 192

Two fundamental quadratic differential forms for a surface 194

Conditions for mutually conjugate or perpendicular tangents . 195

Lines of curvature 196

Dupin's theorem for triply orthogonal systems 197

Curvature of a curve on a surface 199

Dupin's indicatriz 201

Torsion of asymptotic Unes . 202

Total relative curvature, its relation to Gaussian curvature . 20S

Surfaces of zero relative curvature ... ... 204

Surfaces of zero Gaussian curvature 205

Ruled surfaces of zero Graussian curvature in elliptic or spherical

space 206

Geodesic curvature and geodesic lines 208

Necessary conditions for a minimal surface 21Q

Integration of the resulting differential equations .... 212

CHAPTER XVI

DIFFERENTIAL LINE-GEOMETRY

Analoga of Rummer's coefficients . . ... 215

Their fundamental relations 216

Limiting points and focal points 217

Necessary and sufficient conditions for a normal congruence . . 222

Malus-Dupin theorem 225

Isotropic congruences, and congruences of normals to surfaces of

zero curvature 225

Spherical representation of rays in elliptic space .... 227

Representation of normal congruence 228

Isotropic congruence represented by an arbitrary function of the

complex variable 229

Special examples of this representation 232

Study's ray to ray transformation which interchanges parallelism

and parataxy 233

Resulting interchange among the three special types of congruence 235

12 CONTENTS

CHAPTER XVIIMULTIPLY CONNECTED SPACES pôE

Repudiation of the axiom of free mobility of space as a whole . 237

Resulting possibility of one to many correspondence of points andcoordinate sets '237

Multiply connected euclidean planes 239

Multiply connected euclidean spaces, various types of line in them 240

Hyperbolic case little known ; relation to automorphic functions . 243

Non-existence of multiply connected elliptic planes . 245

Multiply connected elliptic spaces 245

CHAPTER XVniPROJECTIVE BASIS OF NON-EUCLIDEAN GEOMETRY

Fundamental notions . . . 247

Axioms of connexion and separation 247

Projective geometry of the plane 249

Projective geometry of space 250

Projective scale and cross ratios 255

Projective coordinates of points in a line 261

Linear transformations of the line 262

Projective coordinates of points in a plane .... 262

Equation of a line, its coordinates 263

Projective coordinates of points in space 264

Equation of a plane 264

CoUineations 265

Imaginary elements 265

Axioms of the congruent collineation group 268

Reappearance of the Absolute and previous metrical formulae . 272

CHAPTER XIXDIFFERENTIAL BASIS FOR EUCLIDEAN AND NON-

EUCLIDEAN GEOMETRYFundamental assumptions 275Coordinate system and distance elements 275

Geodesic curves, their differential equations 277

Determination of a geodesic by a point and direction cosines oftangent thereat 278

Determination of a geodesic by two near points .... 278Definition of angle 279Axiom of congruent transformations 279Simplified expression for distance element 280Constant curvature of geodesic surfaces 281

Introduction ofnew coordinates; integration ofequations ofgeodesic 284Reappearance of familiar distance formulae 284Recapitulation . . 285

Index 287

CHAPTER I

FOUNDATION FOR METRICAL GEOMETRYIN A LIMITED REGION

In any system of geometry we must begin by assumingthe existence of certain fundamental objects, the raw material

with which we are to work. What names we choose to

attach to these objects is obviously a question quite apart

fi-om the nature of the logical connexions which arise fromthe various relations assumed to exist among them, and in

choosing these names we are guided principally by tradition,

and by a desire to make our mathematical edifice as well

adapted as possible to the needs of practical life. In the

present work we shall assume the existence of two sorts

of objects, called respectively points and distances.* Ourexplicit assumptions shall be as follows :

—

« There is no logical or mathematical reason why the point should be takenas undefined rather than the line or plane. This is, however, the invariable

custom in works on the foundations of geometry, and, considering theweight of historical and psychological tradition in its favour, the point

will probably continue to stand among the fundamental indefinables. Withregard to the others, there is no such unanimity. Veronese, Fondammti di

gannetria, Fadua, 1891, takes the line, segment, and congruence of segments.

Schur, 'Ueber die Grundlagen der Oeometrie,' MathematisiAe Annalen, vol.

Iv, 1902, uses segment and motion. Hilbert, IHe Grundlagen der Geometrie,

Leipzig, 1899, uses praotically the same indefinables as Veronese. Moore,' The projective Axioms of Gteometry,' Transactions of the American Mathematical

Society, voL iii, 1902, and Veblen, 'A System of Axioms for Geometry,' sameJournal, vol. v, 1904, use segment and order. Fieri, 'DeUa geometria

elementare come sistema ipotetico deduttivo,' Memorie delta S. Accademia delle

Scieme di Torino, Serie 2, vol. xlix, 1899, introduces motion alone, as does

Fadoa, ' Un nuovo sistema di definizioni per la geometria euclidea,' Periodica

di matematica, Serie 3, vol. i, 1903. Yahlen, Abstrdkte Geomarie, Leipzig, 1905,

uses line and separation. Peano, ' La geometria basata aulle idee di punto

e di distanza,' Attt detta B. Accademia di Torino, vol. xxxviii, 1902-3, andLevy, 'I fondamenti della geometria metrica-proiettiva,' Memorie Accad.

Torino, Serie 2, vol. liv, 1904, use distance. I have made the same choice as

the last-named authors, as it seemed to me to give the best approach to the

problem in hand. I cannot but feel that the choice of segment or order

would be a mistake for our present purpose, in spite of the very condensed

system of axioms which Veblen has set up therefor. For to reach con-

gruence and measurement by this means, one is obliged to introduce the

six-parameter group of motions (as in Ch. XVIII of this work), i. e. base

metrical geometry on projective. It is, on the other hand, an inelegance to

base projective geometry on a non-projective conception such as ' between-

14 FOUNDATION FOR METRICAL GEOMETRY ch.

Axiom I. There exists a class of objects, contaming at

least two members, called points.

It will be convenient to indicate points by large Romanletters as A, B, C.

Axiom II. The existence of any two points implies the

existence of a unique object called their distance.

If the points be A and B it will be convenient to indicate

theii- distance by AB or BA. We shall speak of this also

as the distance between the two points, or from one to the

other.

We next assume that between two distances there mayexist a relation expressed by saying that the one is congruentto the other. In place of the words ' is congruent to ' weshall write the symbol = . The following assumptions shall

be made with regard to the congruent relation :

—

Axiom HI. AB = AB.

Axiom IV. AA = BB.

Axiom V. JI AB = CD and CD = EF, then AB = EF.

These might have been put into purely logical form bysaying that we assumed that every distance was congruentto itself, that the distances of any two pairs of identicalpoints ai-e congruent, and that the congruent relation is

transitive.

Let us next assume that there may exist a triadic relationconnecting three distances which is expressed by a saying

that the first AB is congruent to the sum of the second CDand the third JPQ. This shall be written AB = CD+ FQ.

Axiom VI. if AB = GD + PQ, then AB = PQ + CD.

Axiom VIL 1XAB = GD+PQ and PQ = RS, then

AB = CD+MB.

Axiom VIIL if AB = GD+FQ and A'B' = AB, then

A^=CD + PQ.

Axiom IX. AB = AB + CG.

Definition. The distance of two identical points shall becalled a nidi distance.

ness', whereas writers like Vahlen require both projective and 'affine'geometry, before reaching metrical geometry, a very roundabout way toreach what is, after all, the fundamental part of the subject.

I IN A LIMITED REGION 15

Defirntion. If AB and CD be two such distances that there

exists a not null distance PQ fulfilling the condition that ABis congruent to the sum of CD and PQ, then AB shall be said

to be greater than CD. This is written AB > CD.

Definition. If AB > CD, then CD shall be said to be less

than AB. This is written CD < AB.

Axiom X. Between any two distances AB and CD there

exists one, and only one, of the three relations

AB = CD, AB>CD, AB <CD.

Tlieorem 1. IS AB = CD, then CD = AB.For we could not have AB = CD +PQ where PQ was

not null. Nor could we have CD = AB +PQ for then, by

VIII, AB = AB +PQ contrary to X.

Theorem 2. If AB = CD +PQ and CD' = CD, then

AB = CW+PQ.The proof is immediate.

Axiom XI. If A and C be any two points there exists

such a point B distinct firom either that

AB = AG+ GB.

This axiom is highly significant. In the first place it

clearly involves the existence of an infinite number of points.

In the second it removes the possibility of a maximum dis-

tance. In other words, there is no distance which may not

be extended in either direction. It is, however, fundamentally

important to notice that we have made no assumption as

to the magnitude of the amount by which a distance maybe so extended; we have merely premised the existence of

such extension. We shall make the concept of extension

more explicit by the following definitions.

Definition. The assemblage of all points C possessing the

property that AB = AC+CB shall be called the segment of

A and B, or of B and A, and written (AB) or (BA). Thepoints A and B shall be called the &t;tremities of the segment,

all other points thereof shall be said to be within it.

Definition. The assemblage of all points B different from

A and C such that AB =AC+CB shall be called the extension

of (AC) beyond C


Axiom XII. it AB = lG+GB where AC = AD +W,then AB = AD +DBMrheTeDB = DC+GB.The effect of this axiom is to establish a serial order among

the points of a segment and its extensions, as will be seen

from the following theorems. We shall also be able to showthat our distances are scalar magnitudes, and that addition of

distances is associative.

Axiom XIII. If AB_= PQ +RS there is a single point

C of (AB) such that AG = PQ, GB = RS.

Theorem 3. li AB> CD and CD >EF, then AB>EF^_^To begin with AB = EF is impossible. If then EF>AB,

let us put .EF =M+ GF, where EG = AB.

Then CD = CH + Hb; CB = EF.

Then CD = CK + KD; CK = ABwhich is against our hypothesis.

We see as a corolliuy, to this, that if C and D be any twopoints of{AB), one at least being within it, AB > CD.

It will follow from XIII that two distinct points of asegment cannot determine congruent distances from either endthereof. We also see from Xn that if C be a point of (AB),and D a point of (AC), it is likewise a point of (AB). Letthe reader show further that every point of a segment, whoseextremities belong to a given segment, is, itself, a point ofthat segment.

Theorem 4. If be a point of (AB), then every point D of(AB) isjiithera point of (AC) or of (GB).

If AC = AD we have C and D identical. If IC > AD wemay find a point of (AG) [and so of (.^.5)] whose distance from

A is congruent to AD, and this will be identical with D. If

AC < AD we find C7 as a point of (AD), and hence, by XII,D is a point of (GB).

Theorem 5. If_l5 =AC+GB and Z5 = AD +DB while

AG > AD, then GB < DB.

Theorem 6. If AB = FQ+RS and AW= PQ +M, then

A^ = AB.The proof is left to the reader.

Theorem 7. li IS = FQ + RS and lB = FQ + LM, then

R8 = LM.

I m A LIMITED REGION 17

For if 15 = AG+CB, and AC = Pq, then CB = R8 = LM.

If AB = PQ+RSit will be convenient to write

Pq = (AB-RS),

and say that PQ is the difference of the distances AB and RS.

When we are uncertain as to whether AB > B,8 or MS > AB,we shall write their difference I AB—RS\.

Theorem 8. If 25 = PQ +LM and AB = P^+I/Wwhile PQ = PW,then LM = I/W.

Theorem 9. If AB = PQ + RS and AB = P'Q' + JB'^

while PQ > WQ',

then RSkRW.Definition. The assemblage of all points of a segment and

its extensions shall be called a line.

Definition. Two lines having in common a single point are

said to cut or intersect in that point.

Notice that we have not as yet assumed the existence of

two such lines. We shall soon, however, make this assumptionexplicitly.

Axiom XIV. Two lines having two common distinct points

are identical.

The line determined by two points A and B shall be written

AB or BA.

Theorem 10. If C be a point of the extension of (-^-6)

beyond B and D another point of this same extension, then Dis a point of (BC) it BC = BD or BC >BD; otherwise C is

a point of (BD).

Axiom XV. All points do not lie in one line.

Axiom XYI. if £ be a point of (CD) and E a point of

(AB) where A is not a point of the line BG, then the line DEcontains a point F of (AC).

The first of these axioms is clearly nothing but an existence

theorem. The second specifies certain conditions under whichtwo lines, not given by means of common points, must, never-

theless, intersect. It is clear that some such assumption is

necessary in order to proceed beyond the geometry of a single

straight line.


Theorem 11. If two distinct points A and B be given, there

is an infinite number of distinct points which belong to their

segment.

This theorem is an immediate consequence of the last twoaxioms. It may be interpreted otherwise by saying that there

is no minimum distance, other than the null distance.

Theorem 12. The mainfold of all points of a segment is

dense.

Theorem, 13. If A, B, C, D, E form the configuration of

points described in Axiom XVI, the point^ is a point of {DF).Suppose that this were not the case. We should either

have F aa & point of {DE) or Z) as a point of (EF). But then,

in the first case, C would be a point of (BB) and in the secondD would be a point of (BG), both of which are inconsistent

with our data.

Definition. Points which belong to the same line shall besaid to be on it or to be cdlinear. Lines which contain thesame point shall be said to pass through it, or to be con-current.

Theorem 14. 1{ A, B, C he three non-collinear points, and Da point within (AB) while E ia & point of the extension of(BC) beyond C, then the line BE will contain a point Fof (AC).Take G, a point of (ED), difierent from E and D. Then AG

will contain a point L of (BE), while G belongs to (AL). If Land C be identical, G will be the point required. If L bea point of (CE) then EG goes through F within (AG) asrequired. K i be within (BC), then BG goes through H of(AC) and K of (AE), so that, by 13, G and ff are pointsof (BK). H must then, by 4, either be a point of (BG) or of

(GIf). But if J7 be a point of (BG), C is a point of (BL),which is untrue. Hence f is a point of (GK), and (AH)contains F of (EG). We see also that it is impossible that Gshould belong to (AF) or A to (FC). Hence F belongsto (AC).

Theorem 15. If A, B, C be three non-collinear points, nothree points, one within each of their three segments, arecollinear.

The proof is left to the reader.

Definition. If three non-collinear points be given, the locusof all points of all segments determined by each of these, andall points of the segment of the other two, shall be calleda Triangle. The points originally chosen shall be called the


vertices, their segments the sides. Any point of the triangle,

not on one of its sides, shall be said to be vMhin it. If thethree given points be A, B, C their triangle shall be -written

AABC. Let the reader show that this triangle is completelydetermined by all points of all segments having A as oneextremity, while the other belongs to {BG).

It is interesting to notice that XYI, and 13 and 14, may besummed up as foUows * :

—

Theorem, 16. If a line contain a point of one side of atriangle and one of either extension of a second side, it will

contain a point of the third side.

Definition. The assemblage of all points of all lines deter-

mined by the vertices of a triangle and all points of the

opposite sides shall be called a plav>e.

It should be noticed that in defining a plane in this manner,the vertices of the triangle play a special rdle. It is our next

task to show that this specialization of function is onlyapparent, and that any other three non-coUinear points of the

plane might equally weU have been chosen to define it.t

Theorem 17. If a plane be determined by the vertices of a

triangle, the following points lie therein :

—

(a) All points of every line determined by a vertex, anda point of the line of the other two vertices.

(b) All points of every line which contains a point of each

of two sides of the triangle.

(c) AU points of every line containing a point of one side

of the triangle and a point of the line of another side.

(d) All points of every line which contains a point of the

line of each of two sides.

The proof will come at once from 16, and from the con-

sideration that if we know two points of a line, every other

point thereof is either a point of their segment, or of one of its

extensions. The plane determined by three points as A, B, Gshall be written the plane ABC. We are thus led to the

following theorem.

Theorem 18. The plane determined by three vertices of a

triangle is identical with that determined by two of their

number and any other point of the line of either of the

remaining sides.

* Some writers, as Fasch, Neuire Oeomelrit, Leipzig, 1882, p. 21, giye AxioiaXVI in this form, I have followed Veblen, loc, cit., p. 851, in weakening theaxiom to the form given.

f The treatment of the plane and space which constitute the rest of this

chapter are taken largely from Schur, loc. eit. He in turn confesses his

indebtedness to Peano.

b2


Theorem 19. Any one of the three points determining a plane

may be replaced by any other point of the plane, not comnearwith the two remaining determining points.

Theorem 20. A plane may be determined by any three of

its points which are not colllnear.

Theorem 21. Two planes having three non-collinear points

in common are identical.

Theorem 22. If two points of a line lie in a plane, all points

thereof lie in that plane.

Axiom XYII. All points do not lie in one plane.

Definition. Points or lines which lie in the same plane shall

be called coplanar. Planes which include the same line shall

be called coaxal. Planes, like lines, which include the samepoint, shall be called concurrent.

Definition. K four non-coplanar points be given, the assem-

blage of all points of all segments having for one extremity

one of these points, and for the other, a point of the triangle

of the other three, shall be called a tetrahedron. The four

given points shall be called its vertices, their six segments its

edges, and the four triangles its faces. Edges having nocommon vertex shall be called opposite. Let the reader showthat, as a matter of fact, the tetnihedron will be determinedcompletely by means of segments, all having a commonextremity at one vertex, while the other extremity is in the

face of the other three vertices. A vertex may also be said

to be opposite to a face, if it do not lie in that race.

Defi/nition. The assemblage of all points of all lines whichcontain either a vertex of a tetrahedron, and a point of theopposite face, or two points of two opposite edges, shall becalled a space.

It will be seen that a space, as so defined, is made up offifteen regions, described as follows :

—

(a) The tetrahedron itself.

(b) Four regions composed of the extensions beyond eachvertex of segments having one extremity there, and the otherextremity in the opposite face.

(c) Four regions composed of the other extensions of thesegments mentioned in (b).

(d) Six regions composed of the extensions of segmentswhose extremities are points of opposite edges.

Theorem 23. All points of each of the following figures


will lie in the space defined by the vertices of a giventetrahedron.

(a) A plane containing an edge, and a point of the oppositeedge.

(6) A line containing a vertex, and a point of the planeof the opposite face.

(c) A line containing a point of one edge, and a point of the

line of the opposite edge.

{d) A line containing a point of the line of each of twoopposite edges.

(e) A line containing a point of one edge, and a point of the

plane of a face not containing that edge.

(/) A line containing a point of the line of one edge, anda point of the plane of a face not containing that edge.

The proof wiU come directly if we take the steps in the

order indicated, and hold fast to 16, and the definitions of

line, plane, and space.

Theorem 24. In determining a space, any vertex of a tetra-

hedron may be replaced by any other point, not a vertex, onthe line of an edge through the given vertex.

Theorem 25. In determining a space, any vertex of a tetra-

hedron may be replaced by any point of that space, notcoplanar with the other three vertices.

Theorem, 26. A space may be determined by any four of its

points which are not coplanar.

Theorem, 27. Two spaces which have four non-coplanar

points in common are identical.

Theorem, 28. A space contains wholly every line whereof it

contains two distinct points.

Theorem, 29. A space contains wholly every plane whereofit contains three non-collinear points.

Practical limitation. Points belonging to different spaces

shall not be considered simultaneously in the present work.*

Suppose that we have a plane containing the point E of the

segment (AB) but no point of the segment (BC). Take F andG two other points of the plane, not collinear with E, andconstruct the including space by means of the tetrahedi-on

whose vertices are A, B, F, O. As G lies in this space, it

must lie in one of the fifteen regions individualized by the

* This means, of course, that we shall not consider geometry of more than

three (Umensions. It would not, however, strictly speaking, be accurate to

say that we consider the geometry of a single space only, for we shall malc»

various mutually contradictory hypotheses about space.

22 FOUNDATION FOR METRICAL GEOMETRY ch. I

tetrahedron ; or, more specifically, it must lie in a plane con-

taining one edge, and a point of the opposite edge. Everysuch plane will contain a line of the plane EFO, as may beimmediately proved, and 16 will show that in every case this

plane must contain either a point of {AC) or one of {BC).

Theorem, 30. If a plane contain a point of one side of a

triangle, but no point of a second side, it must contain a point

of the third.

Theorem 31. If a line in the plane of a triangle contain

a point of one side of the triangle and no point of a secondside, it must contain a point of the third side.

Definition. If a point within the segment of two givenpoints be in a given plane, those points shall be said to beon opposite sides of the plane ; otherwise, they shall be said to

be on the same side of the plane. Similarly, we may define

opposite sides of a line.

Theorem 32. If two points be on the same side of a plane,

a point opposite to one is on the same side as the other ; andif two points be on the same side, a point opposite to one is

opposite to both.

The proof comes at once from 30.

Theorem 33. If two planes have a common point they havea common line.

Let P be the common point. In the first plane take a linethrough F. K this be also a line of the second plane, thetheorem is proved. If not, we may take two points of thisline on opposite sides of the second plane. Now any otherpoint of the first plane, not collinear with the three alreadychosen, will be opposite to one of the last two points, and thusdetermine another line of the first plane which intersects thesecond one. We hereby reach a second point common tothe two planes, and the line connecting the two is commonto both.

It is immediately evident that all points common to thetwo planes lie in this line.

CHAPTER II

CONGRUENT TRANSFORMATIONS

In Chapter I we laid the foundation for the present work.We made a number of explicit assumptions, ajid, buildingthereon, we constructed that three-^mensional type of

space wherewith we shall, from now on, be occumed. Anessential point in our system of axioms is this. We havetaken as a fundamental indefinable, distance, and this, beingsubject to the categories greater and less, is a magnitude.In other words, we have laid the basis for a metrical geometry.Yet, the principal use that we have made of these metricalassumptions, has been to prove a number of descriptive

theorems. In order to complete our metrical system properlywe shall need two more assumptions, the one to give us theconcept of continuity, the other to establish the possibility of

congruent transformations.

Axiom XVIIL If all points of a segment (^1^) bedivided into two such classes that no point of the first

shall be at a greater distance from A than is any pointof the second; then there exists such a point C of thesegment, that no point of the first class is within (CB) andnone of the second within (AC)^

It is manifest that A will belong to the first class, and B to

the second, while C may be ascribed to either. It is the

presence of this point common to both, that makes it

advisable to describe the two classes in a negative, rather

than in a positive manner.

Theorem 1. If AB and PQ be any two distances whereof

the second is not null, there wiU exist in the segment (AB)a finite or null number n of points Pj^ possessing the following

properties

:

PQ = AP, = P^;p;~^,; AP^^ÂPj^^T^P^^; P^kPQ.Suppose, firstly, that AB < PQ then, clearly, ti = 0. If,

however, AB = PQ then n = 1 and P^ is identical with B.

There remains the third case where AB > PQ. Imagine the

theorem to be untrue. We shall arrive at a contradiction as

follows. Let us divide all points of the segment into two

24 CONGRUENT TRANSFORMATIONS ch.

classes. A point H shall belong to the first class if we mayfind such a positive integer n that

pjSkpq, ah = ap„+p:s,the succession of points Pj being taken as above. All other

points of the segment shall 'be assigned to the second class. It

is dear that neither class will be empty. If jET be a point

of the first class, and K one of the second, we cannot have

K within (AH), for then we should find AK = AP„ + PjS;P„K < PQ contrary to the rule of dichotomy. We have

therefore a cut of the type demanded by Axiom XVIII, anda point of division G. Let B be such a point of (AG) that

BG< PQ. Then, as we may find n so large that P„D < PQ,we shall either have P„C< PQ or else we shall be able to

insert a point P„+j within (AG) making P„+i(7<PQ. If,

then, in the first case we construct P„+i, or in the second

P„+2, it will be a point within (GB), as P„B>PQ, and this

involves a contradiction, for it would require P„+i or P„+2to belong to both classes at once. The theorem is thus

proved.

It will be seen that this theorem is merely a variation ofthe axiom ofArchimedes,*which says,in non-technical language,

that if a sufficient number of equal lengths be laid off on aline, any point of that line may be surpassed. We are notable to state the principle in exactly this form, however, for

we cannot be sure that our space shall include points of thetype P„ in the extension of (AB) beyond B.

Theorem, 2. In any segment there is a single point whosedistances from the extremities are congruent.


The point so found shall be called the middle point of the

* A good deal of attention has been given in recent years to this axiom.For an account of the connexion of Archimedes' axiom with the continuityof the scale, see Stolz, 'Ueber das Axiom des Archimedes,' JtathemaHscheAnnalen, vol. zxxix, 1891. Halsted, Baiiimal Oeametry (New York, 1901), hasshown that a good deal of the subject of elementary geometry can be builtup without the Archimedian assumption, which accounts for the other-wise somewhat obscure title of his book. Hilbert, loc. cit., Ch. IV, wasthe first writer to set up the theory of area independent of continuity,and Vahlen has shown, loc. cit., pp. 297-8, that volumes may be similarlyhandled. These questions are of primary importance in any work that dealsprincipally with the significance and independence of the axioms. In ourpresent work we shall leave non-arcbimedian or discontinuous geometriesentirely aside, and that for the reason -that their analytic treatment involveseither a mutilation of the number scale, or an adjunction of transfiniteelements thereto. We shall, in fact, make use of our axiom of continuityXVIII wherever, and whenever, it is convenient to do so.

II CONGRUENT TRANSFORMATIONS 25

segment. It will follow at once that if k be any positiveinteger, we may find a set of points PiP2...Pj'_i of thesegment {AB) possessing the following properties

ip,=pjpj;[=F^:^; jpT^.îrj+pjpT^.

We may express the relation of any one of these congruent

distances to AB by writing -Pj-Pj+i = 51 ^^•

Theorem 3. If a not null distance AB be given and apositive integer m, it is possible to find m distinct points of

the segment {AB) possessing the properties

AP, = FjPj,,; APj,, = APj + PjPj,,.

It is merely necessary to take k so that 2*^ >m+ 1 and

find APi = ÂB.

Theorem 4. When any segment {AB) and a positive integer

n are given, there exist «— 1 points DiD^.-.D^.i of the

segment {AB) such that

AD^^DjDjZ.^DÎ^B; ADj;, = ADj + DjDj:^^.

If the distance AB be null, the theorem is trivial. Other-wise^ suppose it to be untrue. Let us divide the points of

{AB) into two classes according to the following scheme.

A point Pi shall belong to the fii'st class if we may construct

)t congi'uent distances according to the method already

illustrated, reaching such a point P„ of {AB) that P„B > AP^ ;

all other points of {AB) shall be assigned to the second class.

B will clearly be a point of the second class, but every pointof {AB) at a lesser distance from A than a point of the first

class, will itself be a point of the first class. We have thus

once more a cut as demanded by Axiom XVIII, and a point

of division D^ ; and this point is different from A.

Let us next assume that the number of successive distances

-congruent to AD^ which, by 1, may be marked in {AB), is k,

and let B]^ be the last extremity of the resulting segments,

so that Z)fc-B < ADi- Let i)fc_i be the other extremity of this

last segment. Suppose, first, that k<n. Let PQ be such

a distance that AI\>PQ > D^B. Let Pi be such a point of

{ADi) that iPi > PQ, kPiDi < PQ-Dj^. Then, by mark-

ing k successive distances by our previous device, we reach

26 CONGRUENT TRANSFORMATIONS CH.

Pj such a point of (ADji) that

P^<B^+(PQ~D^)<TQ<APi.But this is a contradiction, for k is at most equal to n—1,and as P, is a point of the first class, there should be at least

one more point of division Pfc+j. Hence k^n. But k>nleads to a similar contradiction. For we might then find Q,

of the second class so that {k-2)Dî-i) that Oi-z-Djt-i > i Â- Hence,

Q^;:;Dt>iAD,+AD,>AQ„and we may find a (A— l)th point Q^.j. But k—1^ n andthis leads us to a contradiction with the assumption that

Qi should be a point of the second class ; i.e.k = n. Lastly,

we shall find that Dj and B are identical. For otherwise

we might find Qi of the second class so that nDiQi<D„Band mai'king n successive congruent distances reach Q„ within

(i>„5), impossible when Qj belongs to class two. Our theorem

is thus entirely proved, and 2>i is the point sought.

It will be convenient to write AD, = — AB.* n

Theorem 5. If AB and PQ be given, whereof the latter is

not null, we may find n so great that - AB < PQ.


We are at last in a position to introduce the concept of

number into our scale of distance magnitudes. Let AB and PQbe two distances, whereof the latter is not null. It may be

possible to find such a distance BS that qBS=PQ ; pR8= AB.

In this case the number - shall be called the mumerical, q__

vieasure of AB in terms of PQ, or, more simply the Tneasure.

It is clear that this measure may be equally well written

p np ——— or ^^ • There may, however, be no such distance as MIS.q nq j' '

Then, whatever positive integer q may be, we may find LM so

that qLM = PQ, and p so that LM>(AB-pLM). By this

process we have defined a cut in our number system of such

V © +

1

a nature that - and appear in the lower and upper


divisions respectively. If - be a number of the lower, andp' + l ?..—;— one of the upper division, we shall see at once by

• © 7/+

1

reducing to a lowest common denominator that - < , •

Every rational number will feill into the one or the otherdivision. Lastly there is no largest number in the lower

division nor smallest in the upper. For suppose that - is the

largest number of the lower division. Then if

LM > (iB-pIM),

we may find n so large that - LM < (AB—pLMJ. Let us

put ZfjJfj = - LM. At the same time as PQ = nqL^M^ we

may, by 1, find k so large that L-^My > {AB—{np+k) L^M^).

Under these circumstances -^ is a number of the lower

division, yet larger than - • In the same way we may prove

that there is no smallest number in the upper. We havetherefore defined a unique irrational number, and this may be

taken as the measure of AB in terms of PQ.

Suppose, conversely, that - is any rational fraction, and

there exists such a distance AB^ that qAB^> pPQ. Then in

(AB') we may find such a point B that AB =- PQ,i.e. there

will exist a distance having the measure - in terms ofPQ. Next

let r be any irrational number, and let there be such a number©+

1

"in the corresponding upper division of the rational

number system that a distance qAff > ((p + l)PQ) may be

found. Then the cut in the number system will give us a cut

in the segment (AB'), as demanded by XVIII, and a point of

division B. The numerical measure of AB in terms of PQwill clearly be r.

Theorem 6. If two distances, whereof the second is not null,

be given, there exists a unique numerical measure for the first

in terms of the second, and if a distance be given, and there

exist a distance having a given numerical measure in terms


thereof, there will exist a distance having any chosen smaller

numerical measure.

Theorem 7. K two distances be congruent, their measures

in teiTDS ofany third distance are equal.

It will occasionaUy be convenient to write the measure of PQin the form mPQ.

Theorem, 8. li r > n and if distances rPQ and nPQ exist,

then rPQ > nPQ.When m and n are both rational, this comes immediately by

reducing to a common denominator. When one or both of

these numbers is in-ational, we may find a number in the

lower class of the larger which is larger than one in the upperclass of the smaller, and then apply I, 3.

Theorem 9. If AB > CD, the measure of AB in terms of

any chosen not null distance is gi-eater than that of CD in

terms of the same distance.

This comes at once by reduction ad absurdum.It will hereafter be convenient to apply the categories,

congruent greater and less, to segments, when these applyrespectively to the distances of their extremities. We maysimilarly speak of the measure of a segment in tei-ms of

another one. Let us notice that in combining segments or

distances, the associative, commutative, and distributive lawsof multiplication hold good ; e. g.

rnPQ=n-rTQ = mPq, n{AB + OD) =nAB+nCI).Notice, in particular, that the measure of a sum is the sum ofthe measures.

Definition. The assemblage of all points of a segment, or ofaU possible extensions beyond one extremity, shall be called

a hidf-line. The other extremity of the segment shall becalled the bound of the half-line. A half-line bounded by Aand including a point B shall be written

|AB. Notice that

every point of a Ime is the bound of two half-lines thereof.

Definition. A relation between two sets of points (P) and(Q) such that there is a one to one correspondence of distinctpoints, and the distances of corresponding pairs of points arein every case congruent, while the sum of two distances is

carried into a congruent sum, is called a congrueid trans-formation. Notice that, by V, the assemblage of all congruenttransformations form a group. I^ further, a congruenttransformation be possible (P) to (Q), and there be two sets

of points (P') and (Q') such that a congruent transformation


is possible from the set (P) (P') to the set (Q) (Qf), then weshall say that the congruent transformation &om (P) to (Q)has been enlarged to indvde the ads (P') avd (Q').

It is evident that a congruent transformation will carrypoints of a segment^ line, or half-line, into points of a segment,line, or half-line respectively. It will also carry coplanarpoints into coplanar points, and be, in fact, a coUineation,

or linear transformation as defined geometrically. In theeighteenth chapter of the present work we shall see how theproperties of congruent figures may be reached by defining

congruent transformations as a certain six-parameter collinea-

tion group.

Axiom XIX. If a congruent transformation exist betweentwo sets of points, to each half-line bounded by a pointof one set may be made to correspond a half-line boundedby the corresponding point of the other set, in such wise that

the transformation may be enlarged to include all pointsof these two half-lines at congruent distances from their

respective bounds.*

Theorem 10. If a congruent transformation carry two chosenpoints into two other chosen points, it may be enlarged to

include all points of their segments.

Theorem 11. If a congruent transformation carry three

non-collinear points into three other such points, it may beenlarged to include all points of their respective ti-iangles.

Theorevn 12. If a congruent transformation cany four non-coplanar points into four other such points, it may be enlargedto include all points of their respective tetrahedra.

Definition. Two figures which correspond in a congruenttransformation shall be said to be congruent.We shall assume hereafter that every congruent transforma-

tion with which we deal has been Enlarged to the greatest

possible extent. Under these circumstances :

—

Theorem 13. If two distinct points be invariant under acongruent transformation, the .same is true of all points of

their line.

Theorem 14. If three non-collinear points be invariant

* The idea of enlarging a congruent transformation to include additional

points is due to Fasch, loc. cit. He merely assumes that if any point beadjoined to the one set, a corresponding point may be adjoined to the other.

We have to make a much clumsier assumption, and proceed more circum-spectly, for fear of passing out of our limited region.


under a congruent transformation, the same is true of all

points of tbeir plane.

Theorem 15. If four non-coplanar points be invariant under

a congruent transformation the same is true of all points

of space.

Definition. The assemblage of all points of a plane on one

side of a given line, or on that given line, shall be called

a half-plane. The given line shall be called the bound of

the half-plane. Each line in a plane is thus the bound of twohalf-planes thereof.

Suppose that we have two non-collinear haLf-lines with

a common bound A. Let B and C be two other pulnts of

one-half-line, and B' and C two points of the other. Thenby Ch. I, 16, a half-line bounded by A which contains

a point of {BB') will also contain a point of (CC), and vice

versa. We may thus divide all half-lines of this plane,

bounded by this point, into two classes. The assemblage

of all half-lines which contain points of segments whoseextremities lie severally on the two given half-lines shall

be called the interior angle of, or between, the given half-

lines. The half-lines themselves shall be called the sides

of the angle. If the half-lines be \AB,\AC, their interior

angle may be indicated by 4-BAG or 1^ GAB. The point Ashall be called the vertex of the angle.

D^nition. The assemblage of all half-lines coplanar withtwo given non-collinear half-lines, and bounded by the

common bound of the latter, but not belonging to their

interior angle, shall be called the exterior angle of the twohalf-lines. The definitions for sides and vertex shall be as

before. If no mention be made of the words interior or

exterior we shall understand by the word angle, inferior

angle. Notice that, by our definitions, the sides are a part of

the interior, but not of the exterior angle. Let the reader also

show that if a half-line of an interior angle be taken, the

other half-line, collinear therewith, and having the same boundbelongs to the exterior angle.

De/mition. The assemblage' of all half-lines identical withtwo identical half-lines, shall be called their interior angle.

The given bound shall be the vertex, and the given half-lines

the sides of the angle. This angle shall also be called a mUlangle. The assemblage of all half-lines with this bound, andlying in any chosen plane thi-ough the identical half-lines,

shall be called their exterior angle in this plane. The defini-

tion of sides and vertex shall be as before.

11 CONGRUENT TRANSFORMATIONS 31

Definition. Two collinear, but not identical, half-lines ofcommon bound shall be said to be opposite.

Definition. The assemblage of all half-lines having as boundthe common bound of two opposite half-lines, and lying inany half-plane bounded by the line of the latter, shall becaUed an angle of the^ two half-lines in that plane. Thedefinitions of sides and vertex shall be as usual. We notice

that two opposite half-lines determine two angles in everyplane through their line.

We have thus defined the angles of any two half-lines ofcommon bound. The exterior angle of any two such half-

lines, when there is one, shall be called a re-entrant angle.

Any angle determined by two opposite half-lines shall becalled a straight angle. As, by definition, two half-lines forman angle when, and only when, they have a common bound,we shall in future cease to mention this fact. Two angles

will be congruent, by our definition of congruent figures,

if there exist a congruent transformation of the sides of oneinto the sides of the other, in so far as corresponding distances

actually exist on the corresponding half-lines. Every half-

line of the interior or exterior angle will similarly be carried

into a corresponding half-line, or as much thereof as actually

exists and contains corresponding distances.

Definition. The angles of a triangle shall be those non-re-entrant angles whose vertices are the vertices of the triangle,

and whose sides include the sides of the triangle.

D^nition. The angle between a half-line including oneside of a triangle, and bounded at a chosen vertex, and the

opposite of the other half-line which goes to make the angle

of the triangle at that vertex, shall be called an exterior angleof the triangle. Notice that there are six of these, and that

they are not to be confused with the exterior angles of their

respective sides.

Theorem, 16. If two triangles be so related that the sides of

one are congruent to those of the other, the same holds for the

angles.

This is an immediate result of 11.

The meanings of the words opposite and adjacent as applied

to sides and angles of a triangle are immediately evident, andneed not be defined. There can also be no ambiguity in

speaking of sides inclvding an angle.

Theorem 17. Two triangles are congruent if two sides andthe included angle of one be respectively congruent to twosides and the included angle of the other.


The tiTith of this is at once evident when we recall the

definition of congruent angles, and 12.

Theorem 18. If two sides of a triangle be congruent, the

opposite angles are congruent.

Such a triangle shall, naturally, be called isosceles.

Theorem 19. If three half-lines He m the same half-plane

and have their common bound on the bound of this half-

plane; then one belongs to the interior angle of the other

two.

Let the half-lines be|AB,

\AG,

\AD. Connect £ with H

and K, points of the opposite half-lines bounding this half-

plane. IfI

AC,I

AD contain points of the same two sides

of the triangle BHK the theorem is at once evident; if

one contain a point of (BH) and the other a point of (BK),then B belongs to l^GAD.

Theorem 20. If|AB be a half-line of the interior 4- GAD,

thenI

AC does not belong to the interior ^ BAD.

Definition. Two non-re-entrant angles of the same planewith a common side, but no other common half-lines, shall besaid to be adjacent. The angle bounded by their remainingsides, which includes the common side, shall be called their

sv/m. It is clear that this is, in fact, their logical sum,containing all common points.

Definition. An angle shall be said to be congruent to thesum of two non-re-entrant angles, when it is congi'uent to thesum of two adjacent angles, respectively congruent to them.

Definition. Two angles congruent to two adjacent angleswhose sum is a straight angle shall be said to be supple-mentary. Each shall be called the supplement of the other.

Defi/nition. An angle which is congruent to its supplementshall be called a right angle.

Definition. A triangle, one of whose angles is a right angle,shall be called a right triangle.

Definition. The interior angle formed by two half-lines,

opposite to the half-lines which are the sides of a giveninterior angle, shall be called the vertical of that angle. Thevertical of a straight angle will be the other half-plane,coplanar therewith, and having the same bound.

Theorem 21. If two points be at congruent distances fromtwo points coplanar with them, aJl points of the line of thefirst two are at congruent distances from the latter two.


For we may find a congruent transformation keeping theformer points invariant, while the latter are interchanged.

Theorem 22, If(AA^' be a half-line of the interior

4-BAAi, then we cannot have a congruent transformationkeeping

{

AB invariant and carrying|AA^ into

{AA-^^

We may suppose that A^ and Ai are at congruent distances

from A. Let H be the point of the segment (AÂi) equi-

distant from Aj^ and A^'. We may find a congruent trans-

formation canying AA^HA^' into AA^'HA-^. Let this takethe half-line

{AB into

|AG (in the same plane). Then if

IAAi and

|AA^' be taken sufficiently small, AÂ{ will

meet AB or AG as we see by I, 16. This will involve acontradiction, however, for if D be the intersection, it is easy

to see that we shall have simultaneously DAj^ = I)A{ and

BA-^ > BA^' or BA^ < BAJ, for B is unaltered by the con-

gruent transformation, while A-^ goes into A-^.

There is one case where this reasoning has to be modified,

namely, when|AG and

|AB are opposite half-lines, for here

I. 16 does not hold. Let us notice, however, that we mayenlarge our transformation to include the 4-BAAi and4-BAA^ respectively. If

{

AB-^ and|AG^ be two half-lines

of the first angle,|AC^ being in the interior angle of^ BAB^

,

to them will correspondJAB^ and I AG^, the latter being in

the interior angle of ^-BAB^', while by definition, corre-

sponding half-lines always determine congruent angles with

IAB. If, then, we choose any half-line

{AL of the interior

i.BAA{, it may be shown that we may find twocorresponding half-lines

|AL^

\AL{ so situated that

|il£,

belongs to the interior ^Jb^BAL-^ and ^LÂL is congruentto ^ LALy The proof is tedious, and depends onshowing that as a result of our Axiom XVIII, if in anysegment the points be paired in such a way that the

extremities correspond, and the greater of two distances froman extremity correspond to the greater of the two correspond-

ing distances from the other extremity, then there is oneseu-corresponding point* These corresponding half-lines

being found, we may apply the first part of our proof without

fear of mishap.

Theorem 23. If|ilC be a half-line of the interior 4. BAD,

it is impossible to have 4-BAG and 4- BAD mutually

congruent.

* C£ Enriciues, Qeonuiria pniettiva, Bologna, 1898, p. 80.

COOUDSS C


Theorem. 24 An angle is congruent to its vertical.

We have merely to look at the congruent transformation

interchanging a side of one with a side of the other.

We see as a result of 24 that if a half-line|AB make right

angles with the opposite half-lines|AG,

\AG', the verticals

obtained by extending {AB) beyond A will be right angles

congruent to the other two. We thus have four mutuallycongruent right angles at the point A. Under these circnm-

stances we shall say that they are mutvxdly perpendicidarthere.

Theorem, 25. If two angles of a triangle be congruent, the

triangle is isosceles.


Given two non-re-entrant angles. The fii-st shall be said to

be greater than the second, when it is congruent to the sumof the second, and a not null angle. The second shall underthese circumstances, and these alone, be said to be less thanthe first. As the assemblage of all congruent transformations

is a group, we see that the relations greater than, less than,and congruent when applied to angles are mutually exclusive.

For if we had two angles whereof the first was both greaterthan and less than the second, then we should have an anglethat would be both greater than and less than itself, anabsurd result, as we see from 23. We shall write > in placeof greater than, and < for less than, = means congruence.Two angles between which there exists one of these threerelations shall be said to be comparable. We shall later see

that any two angles are comparable. The reason why weeannot at once proceed to prove this fact, is that, so far,

we are not very clear as to just what can be done with ourcongruent transformations. As for the a priori question ofcomparableness, we have perfectly clear definitions of greaterthan, less than, and equal as applied to infinite assenu>lages,but are entirely in the dark as to whether when two suchassemblages are given, one of these relations must necessarilyhold.*

Theorem 26. An exterior angle of a triangle is comparablewith either of the opposite interior angles.

Let us take the triangle ABG, while D lies on the extensionof (BC) beyond C. Let E be the middle point of (AG) andlet DE_meÂB) in F. li BE > EF &xd G o{ {BE) so

that FE = EO. Then we have ^..BAG congruent to t-EGO

* Cf. Borel, Levant sur la thmrU des foncUont, Paria, 1898, pp. 102-S.

11 CONGRUENT TRANSFORMATIONS 35

and less than 3ÊCD. If DE < EF we have t-BACgreater than an angle congruent to ^.EGD,

Theorem 27. Two angles of a triangle ai'e comparable.For they are comparable to the same exterior angle.

Theorem 28. If in any triangle one angle be greater thana second, the side opposite the first is greater than that

opposite the second.

Evidently these sides cannot be congruent. Let us thenhave the triangle ABG where 7^BAG > 4-BGA. We may,by the definition of congruence, find such a point C, of (BG)

that i^CÂG is congruent to /i-G^GA and hence G-Â = Gfi.

It thus remains to show that AB < (4(7j + C7jS). Were such

not the case, we might find jDj of (AB) so that AD^ = AG^,

and the problem reduces to comparing BG^ and BD^. Nowin ABD^G^ we have ^5Dj(7j the supplement of ^-AD^Gjwhich is congruent to 4- AG.D^ whose supplement is greater

than ^-BG^Bi. We have therefore returned to our original

problem, this time, however, with a smaller triangle. Nowthis reduction process may be continued indefinitely, and if

our original assumption be false, the inequalities must alwayslie the same way. Next notice that, by our axiom of con-tinuity, the points G^ of (BG) must tend to approach a point

C of that segment as a limit, and similarly the points D^ of

(AB) tend to approach a limiting point, D. If two points of

{AB) be taken indefinitely close to D the angle which theydetermine at any point of (BG) other than B will becomeindefinitely small. On the other hand as G^ approaches G,

4~APCi wiU tend to increase, where P is any point of (AB)other than B, in which case the angle is constant. This

shows that G, and by the same reasoning D, cannot be other

than B ; so that the difference between BGi and BDi can bemade as small as we please. But, on the other hand

C^ = AGi = AD[; (AZ-W) = (M,-BCi) = (BDi-W^)

Our theorem comes at once from this contradiction.

Theorem 29. If two sides of a triangle be not congruent,

the angle opposite the greater side is greater than that opposite

the lesser.

Theorem 30. One side of a triangle cannot be greater than

the sum of the other two.

Theorem 31. The diflerence between two sides of a triangle

is less than the third side.

The proofs of these theorems are left to the reader.

C2

36 CONGRUENT TRANSFORMATIONS CH.

Theorem 32. Two distinct lines cannot be coplanar with

a third, and perpendicular to it at the same point.

Suppose, in fact, that we have AC and AD perpendicular to

BR at A. We may assume AB = AS so that by I. 31 ADwill contain a single point E either of (GB) or of {GB). For

definiteness, let E belong to {CE). Then take F on (5C),

which is congruent to {BG), so that BF = BE. Hence

^BBF is congruent to %-BBE and therefore congruent to

4.BBE; which contradicts 23*

Theorem 33. The locus of points in a plane at congruent

distances from two points thereof is the line through the middle

point of their segment perpendicular to their line.

Theorem 34. Two triangles are congruent if a side and twoadjacent angles of one be respectively congruent to a side andtwo adjacent angles of the other.

Theorem 35. Through any point of a given line will pass

one line perpendicular to it lying in any given plane throughthat line.

Let A be the chosen point, and G a point in the plane, not

on the chosen line. Let us take two such points B, B on the

given line, that A is the middle point of (BB) and BB < GB,

BB< GB. If then GB = GB, AC is the line required. If

not, let us suppose that GB > GB. We may make a cutin the points of (GB) according to the following principle.

A point F shaU belong to the first class if no point of thesegment (PB) is at a distance &om B greater than its distance

from B, all other points of (GB) shell belong to the secondclass. It is clear that the requirements of Axiom XYIII are

fulfilled, and we have a point of division D. We could not

have DB < DB, for then we might, by 31, take E a point

of {DC) so very near to D that for all points P of BEPB < PB, and this would be contrary to the law of the cut.

In the same way we could not have DB > DB. Hence AD is

the perpendicular required.

Theorem 36. If a line be perpendicular to two others at

* This is substantially Hilbert's proof, loc. cit., p. 16. It is trulyastonishing how much geometers, ancient and modem, have worried overthis theorem. Euclid puts it as his eleventh axiom that all right anglesare equal. Many modem textbooks prove that all straight angles are equal,

hence right angles are equal, as halves of equal things. This is not usuallysound, for it is not clear by de6nition why a right angle is half a straightangle. Others observe the angle of a fixed and a rotating line, and eitherappeal explicitly to intuition, or to a vague continuity axiom.


their point of intersection, it is perpendicular to every line

in their plane through that point.

The proof given in the usual textbooks will hold.

Theorem 37. All lines perpendicular to a given line at

-a given point are coplanar.

Definition. The plane of all perpendiculars to a line at a

point, shall be said to be perpendicular to that line at that

point.

Theo^'em 38. A congruent transformation which keeps all

points of a line invariant, will transform into itself every planeperpendicular to that line.

It is also clear that the locus of all points at congruent

<listances from two points is a plane.

Theorem, 39. If P be a point within the triangle ABC and

there exist a distance congruent to AB + AG, then

AB+AU>PB+PG.To prove this let BP pass through D of {AG). Then as

AG > AB a distance exists congruent to AB-{-AD, and

AB +AD>BP + PD. _ksAB +AD > PS thê exists a dis-

tance congruent to PD + DG, and hence PB + DG > PG,

BG > PG-PB; AB+ AC > BP+ PC.

Theorem 40. Any two right angles are congruent.

Let these right angles be 4.AOG and Â'O'G'. Wemay assume to be the middle point of (AB) and 0' the

middle point of (A'B'), whei-e OA = O'A'. We may also

suppose that distances exist congruent to AG+GB and to

A^+G^. Then AG > AO and A^' > A^'. Lastly, we

may assume that AG = A'G'. For if we had say, AG > A'C',

we might use oui- cut proceeding in (OG). A point P shall

belong to the first class, if no point of (OP) determines with Aa distance greater then A'C', otherwise it shall belong to the

second class. We find a point of division B, and see at once

that AB = A'G'. Replacing the letter B by C, we have

AJG = £G', AABG congruent to AA'B'G', hence SÂOGcongruent to IÂ'O'C

Theorem 41. There exists a congi-uent transformation caiTy-

ing any segment (AB) into any congruent segment (A'R) andany half-plane bounded by AB into any half-plane bounded

hy A'B.We have merely to find and 0' the middle points o{(AB)


and (^'8') respectively, and G and C on the perpendiculars

to AB and A'Bf, at and 0' so that OG = O'C.

Theorem 42. If|OA be a given half-line, there will exist

in any chosen half-plane bounded by OA a unique half-line

OB making the 4—AOB congruent to any chosen angle.

The proof of this theorem depends immediately upon thepreceding one.

Several results follow from the last four theorems. Tobegin with, any two angles are comparable, as we see at oncefrom 42. We see also that our Axioms III-XIII and XVIII,may be at once translated into the geometry of the angle

if straight and re-entrant angles be excluded. We may thenapply to angles system of measurement entirely analogousto that applied to distances. An angle may be representedunequivocidly by a single number, in terms of any chosennot null angle. We may extend our system of comparison toinclude straight and re-entrant angles as foUows. A straight

angle shall be looked upon as greater than every non-re-entrant

angle, and less than every re-entrant one. Of two re-entrant

angles, that one shall be considered the less, whose corre-

sponding interior angle is the greater. A re-entrant anglewill be the logical sum of two non-re-entrant angles, and shall

have as a measure, the sum of their measures.

We have also found out a good deal about the congruentgroup. The principal facts are as follows :

—

(a) A congruent transformation may be found to carry anypoint into any other point.

(b) A congi-uent transformation may be found to leave anychosen point invariant, and carry any chosen line throughthis point, into any other such line.

(c) A congruent transformation may be found to leaveinvariant any point, and any line through it, but to carryany plane through this line, into any other such plane.

(d) If a point, a line through it, and a plane through theline be invariant, no further infinitesimal congruent trans-formations are possible.

The last assertion has not been proved in full; let thereader show that if a point and a line through it be invariant,there is only one congruent transformation of the line possible,

besides the identical one, and so on. The essential thingis this. We shall demonstrate at length in Gh. XYIII thatthe congruent group is completely determined by the require-ment that it shall be an analytic collineation group, satisfyingthese four requirements.

S'

u CONGRUENT TRANSFORMATIONS 39

Suppose that we have two half-planes on opposite sides

of a plane a which contains their common bound I. Everysegment whose extremities are one in each of these half-planes

will have a point in a, and, in fact, all such points vnll lie

in one half-plane of a bounded by I, as may easily be shownfrom the special case where two segments have a commonextremity.

Definition. Given two non-coplanar half-planes of commonbound. The assemblage of all half-planes with this bound,containing points of segments whose extremities lie severally

in the two given half-planes, shall be called their interior

dihedral angle, or, more simply, their dihedral angle. Theassemblage of all other half-planes with this bound shall becalled their exterior dihedral angle. The two given half-planes

shall be called the faces, and their bound the edge of the

dihedral angle.

We may, by following the analogy of the plane, define null,

straight, and re-entrant dihedral angles. The definition of the

dihedral angles of a tetrahedron will also be immediatelyevident.

A plane perpendicular to the edge of a dihedral angle will

cut the faces in two half-lines perpendicular to the edge.

The interior (exterior) angle of these two shall be called aplane angle of the interior (exterior) dihedral angle.

Theorem 43. Two plane angles of a dihedral angle are con-

gruent.

We have merely to take the congruent transformation

which keeps invariant all points of the plane whose points

are equidistant from the vertices of the plane angles. Sucha transformation may properly be called a reflection in that

plane.

Theorem 44. If two dihedral angles be congruent, any twoof their plane angles will be congruent, and conversely.

The proof is immediate. Let us next notice that we maymeasure any dihedral angle in terms of any other not null one,

and that its measure is the measure of its plane angle in

terms of the plane angle of the latter.

Definition. If the plane angle of a dihedral angle be a right

angle, the dihedral angle itself shall be called righi, and the

planes shall be said to be inutually perpendicular.

Theorem 45. If a plane be perpendicular, to each of twoother planes, and the three be concurrent, then the first

plane is also perpendicular to the line of intersection of the

other two.

CHAPTER III

THE THREE HYPOTHESES

In the last chapter we discussed at some length the problemof comparing distances and angles, and of giving themnumerical measures in terms of known units. We did not

take up the question of the sum of the angles of a triangle,

and that shall be our next task. The axioms so far set upare insufficient to determine whether this sum shall, or shall

not, be congruent to the sum of two right angles, as we shall

amply see by elaborating consistent systems of geometrywhere this sum is greater than, equal to, or less than tworight angles. We must first, however, give one or twotheorems concerning the continuous change of distances andangles.

Theorem 1. If a point P of a segment (AB) may be takenat as small a distance from A as desired, and G be any otherpoint, the 4-ACP may be made less than any given angle.

If C be a point of AB the theorem is trivial. If not, wemay, by III. 4, find

|CD in the half-plane bounded by CA

which contains B, so that 4~ACI) is congruent to the givenangle. If then

{

AB belong to the internal 4-AGJ), we have4ÂCB less than 4.AGD, and, a fortiori, 3ÂGP<4.ACD.If

I

AD belong to the internal 4-AGB,\AD must contain a

point E of CAB, and if we take P within {AE), once more

4-AGP < 4.ACD.

Theorem 2. If, in any triangle, one side and an adjacentangle remain fixed, while the other side including this anglemay be diminished at will, then the external angle oppositeto the fixed side will take and retain a value difiering fromthat of the fixed angle by less than any assigned value.

Let the fixed side be (AB), while G is the variable vertex

within a fixed segment (BD). We wish to show that if 5Cbe taken sufficiently small, ÂGD will necessarily differ from%-ABD by less than any chosen angle.

Let 5i be the middle point of {AB), and B^ the middlepoint of {B^B), while B^ is a point of the extension of {AB)beyond B. Through each of the points By B^, B^ constructa half-line bounded thereby, and lying in that half-plane,

CH. HI THE THREE HYPOTHESES 41

bounded by AB which contains D, and let the angles soformed at Bj, B^) -^s aU ^ congruent to 4—^BD. We maycertainly take BC so small that AG contains a point of eachof these half^hnes, say 0-^,0^, C^ respectively. We may more-

over take BG so tiny that it is possible to extend (B^G^)

beyond C^ to D^ so that B^ Cj = C^Dy AD^ will surely meetB2C2 in a point Dj, when B^G^ is very small, and as AC^differs infinitesimally from AB^, and hence exceeds AB by

a finite amount, it is greater than 2AGi which differs in-

finitesimally from 2ABi, or AB. We may thus find C" on

the extension of (AG^) beyond G^ so that AG^ = G^G'. C will

be at a small distance from G, and hence on the other side of

B^D^ from A and D,. Let DjC' meet B^D^ at H^. We nowsee that, with regard to the A AB-^D^ ; the external angle at

D, (i.e. one of the mutually vertical external anês) is

Îb^D^D^ congruent to {^.B^D^C' + ^-G'D^D^), and ^^^D^G'

42 THE THREE HYPOTHESES OH.

is congruent to^^.^^Dj , and, hence congruent to 4ÂBD. The4-C'D^D^ is the difference between ^-B-^D-yD^ and T^ByD^H^,and as H^ and D^ approach B^ aa & limiting position, the

angles determined by B^, D^ and D^, H^ at every point in

space decrease together towards a null angle as a limit.

Hence ^CD^D.^ becomes infinitesimal, and the difference

between ^B^D^D^ and %ÂBD becomes and remains in-

finitesimal. But as ABi = B^B, and ^--^î^î ^^^ ^B-^BDare congruent, we see similarly that the difference between/^B-^CD and iÂBD will become, and remain infinitesimal.

Lastly, the difference between T^^B^GD and 4-ACD is ^.BfiAwhich will, by our previous reasoning, become infinitesimal

with ^jCj. The difference between 4-ABD and 4-ACD will

therefore become and remain less than any assigned angle.

Several corollaries follow immediately from this theorem.

Theorem 3. If in any triangle one side and an adjacentangle remain fixed, while the other side including this anglebecomes infinitesimal, the sum of the angles of this triangle

will differ infinitesimally from a straight angle.

Theorem 4. If in any triangle one side and an adjacentangle remain fixed, while the other side including this anglevaries, then the measures of the third side, and of the variable

angles will be continuous functions of the measure of thevariable side first mentioned.Of course a constant is here included as a special case of

a continuous function.

Theorem 5. Iftwo lines AB, AG he perpendicular to BC, thenall lines which contain A and points of BG are perpendicularto BG, and all points of BG are at congi-uent distances from A.To prove this let us first notice that our A ABG is isosceles,

and AB will be congruent to every other perpendiculardistance from A to BG. Such a distance will be the distance

from A to tbe middle point of (BG) and, in fact, to everypoint of BG whose distance from B may be expressed in the

form — BG where m and n are integers. Now such points

will lie as close as we please to every point of BG, hence

by U. 31, no distance from A can differ from AB, and noangle so formed can, by m. 2, differ from a right angle.

TheoreTTi 6. If a set of lines perpendicular to a line I, meeta line m, the distances of these points &om a fixed point of m,and the angles so formed with m,, will vary continuously with

Ill THE THREE HYPOTHESES 4»

the distances from a fixed point of 2 to the intersections withthese perpendiculars.

The proof comes easily from 2 and 5.

D^nition. Given four coplanar points A, B, C,D so situated

that no segment may contain points within three of thes^pnents (XB), {BG), (CD), (DA). The assemblage of all pointsof all segments whose extremities lie on these segments shall

be called a quadrilateral. The given points shall be called

its vertices, and the given segments its sides. The fourinternal angles T^-DAB, 4. ABC, 4-BCD, 4.CLA shall becalled its angles. The definitions of opposite sides andopposite vertices are obvious, as are the definitions for

adjacent sides and vertices.

Definition. A quadrilateral with right angles at twoadjacent vertices shall be called birectangular. If it havethree right angles it shall be called trirectangular, and fourright angles it shall be called a rectangle. Let the readerconvince himself that, under our hypotheses, birectangularand trirectangular quadrilaterals necessarily exist.

Definition. A birectangular quadrilateral whose oppositesides adjacent to the right angles are congruent, shall be said

to be isoecelea.

Theorem 7. Saccheri's.* In an isosceles birectangular quad-rilateral a line through the middle point of the side adjacentto both right angles, which is perpendicular to the Une ofthat side, will be perpendicular to the line of the oppositeside and pass through its middle point. The other two angles

of the quadrilateral are mutually congruent.Let the quadrilateral be ABCD, the right angles havii^

their vertices at A and B. Then the perpendicular to ABat E the middle point of {AB) will surely contain F point of

(GD). It will be easy to pass a plane through this line

perpendicular to the plane of the quadrilateral, and by takinga reflection in this latter plane, the quadrilateral will betransformed into itself, the opposite sides being interchanged.

This theorem may be more briefly stated by saying that

* Soccheri, Eudides ab amni naevo vindicaius, Milan, 1732. Accessible in

Engel und Staeckel, ITuorie der PanUlellinieti van Euklid bis aiaf OaaisSf Leipzig,

1895. The theorem given above covers Saccheri's theorems 1 and 2 on p. 60of the last-named work. Saccheri's is the first systematic attempt of whichwe have a record to prove Euclid's parallel postulate, and proceeds accordingto the modem method of assuming the postulate untrue. He builded better

than he knew, however, for the system so constructed is self-consistent, andnot inconsistent, as he attempted to show.

44 THE THREE HYPOTHESES oh.

this line divides the quadrilateral into two mutually congi-uent

trirectangular ones.

Theorem 8. In a rectangle the opposite sides are mutually

congruent, and any isosceles birectangular quadrilateral whose

opposite sides are mutually congruent is necessarily a rectangle.

Theorem 9. If there exist a single rectangle, every isosceles

birectangular quadrilateral is a rectangle.

Let ABCD be the rectangle. The line perpendicular to

AB at the middle point of (AB) will divide it into twosmaller rectangles. Continuing this process we see that wecan construct a rectangle whose adjacent sides may have any

measures that can be indicated in the form -^ AB, ^ AC,

provided, of course, that the distances so called for exist

simultaneously on the sides of a birectangular isosceles

quadrilateral. Distances so indicated will be everywheredense on any line, hence, by 6 we may construct a rectangle

having as one of its sides one of the congruent sides of anyisosceles birectangular quadrilateral, and hence, by a repetition

of the same process, a rectangle which is identical with this

quadrilateral. All isosceles birectangular quadrilaterals, andall trirectangular quadiilaterals are under the present circum-stances rectangles.

Be it noticed that, under the present hypothesis. Theorem 5

is superfluous.

Theorem 10. If there exist a single right triangle the sumof whose angles is congruent to a straight angle, the same is

true of every light triangle.

Let AABC be the given triangle, the right angle being

^ ACB so that the sum of the other two angles is congruentto a right angle. Let AA'B'C be any other right triangle,

the right angle being ^ A'CB'. We have to prove that thesum of its remaining angles also is congruent to a right angle.We see that both ÂBC and ^BAC are less than rightangles, hence there will exist such a point E of {AB) that4-EAC and 4ÊCA are congruent. Then 4.EBC = 4.ECBsince 4- ACB is congruent to the sum of 4-EAC and 4-EBC.If Z) and F be the middle points of (BG) and (AC) respec-tively, as AEAC and AEBG are isosceles, we have, in thequadrilateral EDGF right angles at D, C, and F. The angleat E is also a right angle, for it is one half the straight angle,4-AEB, hence 4-EDGF is a rectangle. Passing now to theAA'G'F we see that the perpendicular to A^' at F' the

Ill THE THREE HYPOTHESES 45

middle point of (A'C), wiU meet (A'B') in E', and the per-pendicular to EF' at E will meet {RC} in IT. But, byan easy modification of 9, as there exists one rectangle, thetrirectangular quadrilateral E'F'B'C is also a rectangle. It

is clear that t-D'E'B =i.D'E'G' since i^F'E'iy is a right

angle and ^.F'E'A' = i^FE'C. Then /^CE'R is isosceles

like Â'E'C. From this comes immediately that the sumof Ê'EG' and "Ê'A'C is congruent to a right angle, as

we wished to show.

Theorem, 11. If there exist any right triangle where the

sum of the angles is less than a straight angle, the same is

true of all right triangles.

We see the truth of this by continuity. For we may pass

from any right triangle to any other by means of a continuous

change of first the one, and then the other of the sides whichinclude the right angle. In this change, by 2, the sum of the

angles will either remain constant, or change continuously,

but may never become congruent to the sum of two right

angles, hence it must always remain less than that sum.

Theorem. 13. If there exist a right triangle where the sumof the angles is greater than two right angles, the same is

true of every right triangle.

This comes immediatdy by reductio ad absurdv/ni.

Theorem, 13. If there exiflt any triangle where the sum ofthe angles is less than (congruent to) a straight angle, then in

every triangle the sum of the angles is less than (congruent

to) a straight angle.

Let us notice, to begin with, that our given AABCmust have at least two angles, say 4-ABG and ^BAC whichare leas than right angles. At each point of {AB) there will

be a perpendicular to AB (in the plane BG). IS two of

these perpendiculars intersect, all will, by 5, pass throughthis point, and a line hence to G will surely be perpendicular

to AB. If no two of the perpendiculars intersect, then,

clearly, some will meet (AG) and some {BG). A cut will

thus be determined among the points of {AB), and, by XYIII,

we shall find a point of division X>. It is at once evident

that the perpendicular to AB at D will pass through G. In

every case we may, therefore, divide our triangle into tworight triangles. In one of these the sum of the angles mustsurely be less than (congruent to) a straight angle, and the

same will hold for every right triangle. Next observe that

there can, under our present circumstances, exist no triangle

with two angles congruent to, or greater than right angles.

46 THE TELREE HYPOTHESES CH.

Hence every triangle can be divided into two right triangles

as we have just done. In each of these triangles, the sum of

the angles is less than (congruent to) a straight angle, hence

in tile triangle chosen, the sum of the angles is less than

(congruent to) a straight angle.

Theorem 14. If there exist any triangle where the sumof the angles is greater than a straight angle, the same will

be true of every triangle.

This comes at once by reductio ad ahsurdum.We have now reached the fundamental fact that the sum of

the angles of a single triangle will deteimine the nature

of the sum of the angles of every triangle. Let us set the

various possible assumptions in evidence.

The assumption that there exists a single triangle, the sumof whose angles is congruent to a straight angle is called the

Eudidean or Parabolic hypothesis.*

The assumption that there exists a triangle, the sum of

whose angles is less than a straight angle is called the

Lobaichewskian or hyperbolic hypothesis.f

The assumption that there exists a triangle, the sum of

whose angles is greater than a straight angle, is called the

Riemanman or aliptic hypothesis. %Only under the elliptic hypothesis can two intersecting

lines be perpendicular to a third line coplanar with them.

Definition. The difference between the sum of the angles of

a triangle, and a straight angle shall be called the discrepancyof the triangle.

Theorem, 15. If in any triangle a line be drawn from onevertex to a point of the opposite side, the sum of the dis-

crepancies of the resulting triangles is congruent to thediscrepancy of the given triangle.

* There will exist, of course, numerous geometries, other than those whichwe give in the following pages, where the sum of the angles of a triangle is

still congruent to a straight angle, e. g. those lacking our strong axiom ofcontinuity. Cf. Dehn, ' Die Legendre'schen Satze uber die Winkelsninme imDreiecke,' MaOtematischt Annalen, toI. liu, 1900, and B. L. Moore, ' (Geometryin which the sum of the angles of a triangle is two right angles,' Transactions

of the American Mathematical Society, voL vlii, 1907.

t The three hypotheses were certainly familiar to Saccheri (loc. cit. ), thoughthe credit for discovering the hyperbolic system is generally given to Gûss,who speaks of it in a letter to Bolyai written in 1799. Lobatchewsky's first

work was published in Russian in Kasan, in 1829. This was followed by anarticle ' 66om€trie imaginaire ', Crdle's Journal, vol. xvii, 1837. All spellingsof Lobatchewsky's name in Latin or Germanic languages are phonetic. Theauthor has seen eight or ten different ones.

X Riemann, Ueber die Hypothesen, welche der Geometrie zu Qrundt liegen, first readin 1854; see p. 272 of the second edition of his Gesammelte Werke, withexplanations in the appendix by Weber.

Ill THE THREE HYPOTHESES 47

The proof is immediate. Notice, hence, that if in anytriangle, one angle remain constant, while one or both of theother vertices tend to approach the vertex of the fixed angle,

along fixed lines, the discrepancy of the triangle, when notzero, will diminish towards zero as a limit. We shall makethis more dear by saying

—

Theorem 16. If, in any triangle, one vertex remain fixed,

the other vertices lying on fixed lines through it, and if asecond vertex may be made to come as near to the fixed vertex

as may be desired, while the third vertex does not tend to

recede indefinitely, then the discrepancy may be made less

than any assigned angle.

Theorem, 17. If in any triangle one side may be made less

than any assigned segment, while neither of the other sides

becomes indefinitely large, the discrepancy may be made less

than any assigned angle.

If neither angle adjacent to the diminishing side tend to

approach a straight angle as a limit, it will remain less thansome non-re-entrant angle, and 16 will apply to all such

angles simultaneously. If it do tend to approach a straight

angle, let the diminishing side be (AB), while 4-BAC tends

to approach a straight angle. Then, as neither BG nor AGbecomes indefinitely great, we see that A must be very close

to some point of the extension of {AB) beyond ^, or to îtself. If G do not approach A, we may apply 1 to show that

jÂGB becomes infinitesimal. If C do approach A we maytake D the middle point of {AC) and extend {BD) to E beyond

D so that DE = EB. Then we may apply Euclid's ownproof* that the exterior angle of a triangle is greater thaneither opposite interior one, so that the exterior angle at Awhich is infinitesimal, is yet greater than 4~ACB.

Theorem 18. If, in any system of triangles, one side of each

may be made less than any assigned segment, all thus

diminishing together, while no side becomes indefinitely

great, the geometry of these triangles may be made to differ

from the geometry of the euclidean hypothesis by as little as

may be desired.

A, specious, if loose, way of stating this theorem is to saythat in the infinitesimal domain, we have euclidean geometry.t

* Euclid, Book I, Proposition 16.

f This theorem, loosely proved, is taken as the basis of a number of vrorks

on non-euclidean geometry, which start in the infinitesimal domain, andwork to the finite by integration. Cf. e. g. Flye Ste-Marie, j^bides analylijutt

surla ihiorit des paraUeles, Paris, 1871.

CHAPTER IV

THE INTRODUCTION OF TRIGONOMETRICFORMULAE

The first fundamental question with which we shall haveto deal in this chapter is the following. Suppose that wehave an isosceles, birectangular quadrilateral ABGD, whose

right angles are at A and B. Suppose, further, that ABbecomes infinitesimally small, AD remaining constant ; what

wiU be the limit of the fraction —^= where M XT means theV.AB

measure of XF in terms of some convenient unit.* But, first

of all, we must convince ourselves, that, when AD is given

we may always construct a suitable quadrilateral ; secondly,

and most important, we must show that a definite limit does

necessarily exist for this ratio, as AB decreases towards the

nuU distance.

Theorem 1. If AD and AX be two mutually perpendicular

lines we may find such a point B on either half ofAX boundedby A, that, a line being drawn perpendicular to AB at anypoint P of {AB) we may find on the half thereof bounded byiP, which lies in the same half-plane bounded by AB as does D,

a point whose distance from P is greater than AD.Let Ehe& point of the extension of {AD) beyond D. Draw

a line there perpendicular to AD. If S be a point of AXvery close to A, and if a line perpendicular to AB at Pof {AB), meet the perpendicular at .E7 at a point Q, PQ differs

but little firom AE, and, hence, is greater than AD.

* The general treatment, and several of the actual proob in this chapterare taken directly from Gerard, La geonulrie non-«ucIiii<mn«, Paris, 1S92. It hasbeen possible to shorten some of his work by the consideration that we haveenclidean geometry in the infinitesimal domain. On the other hand, severalimportant points are omitted by him. There is no proof that the requiredlimit does actually exist, and worse still, he gives no proof that the reanlting

function of uAD is necessarily continuous, thereby rendering valueless hissolution of its functional equation.

OH. IV TRIGONOMETRIC FORMULAE 49

The net result of theorem 1 is this. 1£ AD be given, andthe right 4^DAX, any point of AX very near to A may be

taken as the vertex of a second right angle of an isosceles

birectangular quadrilateral, having A as the vertex of oneright angle, and (AD) as one of the congruent sides.

Definition. We shall say that a distance may be madeinfinitesimal compared with a second distance, if the ratio

of the measure of the first to that of the second may be madeless than any assigned value.

Theorem, 2. If in a triangle whereof one angle is constant,

a second angle may be made as STncdl as desired, the side

opposite this angle will be infinitesimal compared to the other

sides of the triangle.

Suppose that we have, in fact, APQR with 4-PQl^ fixed,

while 4-^^Q becomes infinitesimal. It is clear that one

of the angles 4-PQ^ or 3^QPR must be greater than a right

angle. Suppose it be 4-QP^- Then, by hypothesis, nomatter how large a positive integer n may be, I may find such

positions for P and R, that n points Qi may be found on|

PQ'

BO that 4.PRQ=t-QIiQi = 4-Qh^Qk-n' yet 4-QIlQn is less

than any chosen angle. Now if RQ remain constantly greater

than a given not null distance, the theorem is perfectly

evident. If, on the other hand, RQ decrease indefinitely, we

may find -S on|PQ but not in (PQ), so that QR = QS. Then,

as geometry in the infinitesimal domain obeys the euclidean

hypothesis, 4-Q^^ "^^ differ infinitesimally &om one half

4-PQR- If. then, we require 4-QRQ^ to be less than this last-

named amount, Q„ will be within (QS), and PQ< QjcQu+i

and PQ < - QR. A similar proof holds when 4-PQ^ ^

greater than a right angle.

It will follow, as a corollary, that if in any triangle, one

angle become infinitesimal, and neither of the other angles

approaches a straight angle as a limit, then the side opposite

the infinitesimal angle becomes infinitesimal as comparedwith either of the other sides.

Theorem 3. If in an isosceles birectangular quadrilateral,

the congruent sides remain constant in value, while the side

adjacent to the two right angles decreases indefinitely, the

ratio of the measures of this and the opposite side approaches

a definite limit.

It will save circumlocution and involve no serious confusion

if, during the rest of this chapter, we speak of the ratio of two

50 THE INTRODUCTION OF ch.

distances, instead of the ratio of their measures, and write

PQsuch a ratio simply —— . Let us then take the isosceles

birectangular quadrilateral A'ABB", the right angles havingtheir vertices at A and B. Let us imagine that A and A' are

fixed points, while £ is on a fixed line at a very small distance

from A. Let C be the middle point of (AB), and let the

perpendicular to AB at G meet {A'B') at C, which, bySaccheri's theorem, is the middle point of (A'B'). Now, byin. 6, 4-(^'A'A differs infinitesimaUy from a right angle,

as AG becomes infinitesimal, so that if C^j be the point

of (GC), or {CC) extended beyond C", for which CC^ = 127,

C,C < -AC . But -= =-^= • Hence ? < 6^ n AC AB AG AB

where 8 may be made less than any assigned number. By a re-

peated use ofthis process we see that if -D be such a point oi(AB)Jc

that AB = — AB and Dj such a point of the perpendicular

at D that AA' = BB^, then, however small e may be,

JADl 37B' —-^= ^:^ < e, and, what is more, we may take AB soAJJ AB '

small that this inequality shall hold for all such points D

at once, for, as AB decreases, every ratio ^ gets nearer andaW -^^

nearer to -r=^ • Lastly, ifP be any point of (AB), and P^ lie

on the perpendicular at P so that AA' = PPi, we may find

one of our points recently called B of such a nature that i)Pj

and DjP] are infinitesimal as compared with AB. Hence

^/p 3^-r=? =- < e where € is infinitesimal with AB. ThisAP AB

^'£'shows that approaches a definite limit, as AB approaches

the null distance.

This limit is constantly equal to 1 in the euclidean case.

In the other cases it is a variable depending on the measure

of AA'. If this measure be x, we may call our limit <j> (x).

Let us next show that the function<l>

is continuous. TakeA'ABB' as before, while A^ and B^ are respectively on the

ly TRIGONOMETRIC FORMULAE 51

extensions of {AA'), beyond A', and of (BR) beyond jB'. Let

the measure of AA' be x, while that of A'A^^ is Ax,

Now

A'R

AB

52 THE INTRODXJCTION OF oh.

use letters of the type 8, e, ?/, to indicate infinitesimaJiS, and

remember that AB is an infinitesimal distance.

2^ =I

C;D^-PQ\,_2GJI =I

C:D,-RS\,

GD = <t>(x)AB + eiAB,

a[Di = ^{x-y)AB+iiAB,

C^^ = 4,{x + y)AB+(slB,

PQ = {vlGF)CD + \GT>,

RS ={uCS) CD + h^GD.

But C^ > gU^-'GP and ^P is rnfinitesimal.

'PQ=4,{y)GB+l^GD,

RS = <(>(y)GD + btCD.

Substitute in the first equation connecting G^P and C^R

['i>{x+y) + f^-<t>(x)it>(y)-4,{x)b^-<l>{y)ei + b^fi]yiAB == [<f>(x)<l>{y)+<l>{x)b^+ <t,{y)€i+ btfi-<f>{x-y)-i^] viAB + 2ye.

Hence <f>(x+ y)+ (f>{x—y)—2<l)(x) (f>{y) < rj where »/ may bemade less than any assigned value

<l,{x+y) + <t,{x-y) = Z<l,(x)<l>(y). (1)

This well-known equation may be easily solved. Let usassume that the unit oi measure of distance is well fixed

,^(0)=1, <^(2a;) = 2[<^(a;)p-l.

Let a?! be a value for x in the interval to which the equationapplies, i.e. the measure of an actual distance. We may find

k so that <l»{xj) = cos -r^- We have immediately

We also know that ^(a;)— cos-r is a continuous function.

If, then, X be any value of the argument, we may find n andfixm such large integers that x— ^ is infinitesimal. Hence

X^(«)—COST will be less than any assigned quantity, or

,<,(a;) = cos|. (2)

IV TRIGONOMETRIC FORMULAE 58

The function cosine has, of course, a purely analyticalmeaning, i.e. we write

flu fl3^

Of fundamental importance is the constant k. We shall£nd that it gives the radius of a sphere (in our usualeuclidean geometry) upon which the non-euclidean planemay be developed. We shall, therefore, define the constant

p as the Measure of Curvature of Spcice.* To find the

nature of the value of k, we see immediately that in the

parabolic case p = ; in the elliptic<f>

is, at most, equal

to 1, hence y, is positive. In the hyperbolic case, 1 con-

..' 1

stitutes a minimum value for<f>and ^ is negative, orka, pure

imaginary. Under these circumstances, we may, if we choose,

remove all signs of imaginary values from (2) by writing

JC = tic, /X \4> (x) = cosh [j,)

As a matter of fact, however, there is little or no gain in

doing this.

It is now necessary to calculate another limit, that of the

ratio of two simultaneously diminishing sides of a right

triangle. Let us, then, suppose that we have a right

A ABC whose right angle is ÂBC. We shall imagine that

AB becomes infinitesimal while 4-BAC is constant. WeAB

seek the limit of r^.f That such a limit will actuallyAO

exist may be proved by considerations similar to those whichestablished the existence of 4>{x). We leave the details to

the reader. The limit is a function of the angle l^BAC, andif fl be the measure of the latter, we may write our function

f{d) ; including therein, of course, the possibility that this

function should be a constant.

First of all it is incumbent upon us to show that this

function is continuous. Take C on the extension of {BC)

beyond C, and let Afl be the measure of 4-GAC. If A6 be

* This fundamental concept is due to Riemann, loc, cit. We sliall

consider it more fully in subsequent chapters, notably XIX.

f It is strange that Gerard, loc. cit., assumes this ratio from the euclidean


iniiiiitesimal, then, by 2 GC is infinitesimal as compared

with AG. Hence -=—=- will become and remain less

AB ABthan any assigned number, and f{d) is continuous.

Suppose, now, that we have two half-lines|OF,

|OZ lying

in a half-plane bounded by|OX. Let :I^XOT and 4-XOZ

be each less than a right angle, and have the measures 6, 6 + (!>•,

<t><e. Take F on|OZ, and find B, so that

0^=^;4.Y0F = 4.Y0B,

I

OB is within the interior angle 4-XOY; these points will

certainly exist if Oi^ be very small. Connect F and 5 by a line

meeting|OF in D, and through F, D, B draw three lines per-

pendicular toI

OX, and meeting iiin E,G,A respectively, which

points also are sure to exist, if OF be small enough. G will

be separated from the middle point of {EA) by a distance

infinitesimal compared with EA, for the perpendicular to OXat such a point would meet {BF) at a point whose distance

from D was infinitesimal as compared with OF.

OB OD OB ^ '-^ ^

PA

OE OE ..^ ,,

OB OF ^

^ =f(0)fW-f{e + <l>) + e, •^ -^ = fi. infinitesimal.

f{e + <l>)+f{0-<t>) = 2fie)f{i,).

This is the functional equation that we had before, so that

f = COS J and I must be real. If, then, we so choose it that the

measure of a right angle shall be ^ >

f{0) = cosfl.


Let us not fail to notice that since 4-^BG is a right anglewe have, by in. 17,

lim. == = cos (^ -^ ) = sin 6. (3)

The extension of these functions to angles whose measuresTT

are greater than ^ wîU afford no difficulty, for, on the one

hand, the defining series remains convergent, and, on theother, the geometric extension may be effected as in theelementary books.

Our next task is a most serious and fundamental one, tofind the relations which connect the measures and sides andangles of a ri^ht tiiangle. Let this be the AABC with4-ABC as its right angle. Let the measure of ^BAG be i|r

while that oi^BGA is 0. We shall assume that both i/f and d

TT

are less than »> an obvious necessity under the euclidean

or hyperbolic hypothesis, while under the elliptic, such will

stiU be the case if the sides of the triangle be not large, andthe case where the inequalities do not hold may be easily

treated from the cases where they do. Let us also call a, b, c

the measures of BG, GA, AB respectively.

We now make rather an elaborate construction.* Take Bîn (AB) as near to .8 as desired, and A^ on the extension

of (AB) beyond A, so that AÂ = Bj^B, and construct

AA^B-iCi = AABG, Gi lying not far from C; a construction

which, by 1, is surely possible if BB.^ be small enough. Let

5iCi meet [AG) at G^. t-G^G^G will differ but little fromj^BGA, and we may draw GÔ^ perpendicular to CCj- whereC^ is a point of (GG^. Let us next find A^ on the extension

of (AG) beyond A so that AÂ = G^G and B^ on the extension

of (GiB^) beyond B^ so that B^B^ = GiG^, which is certainly

possible as CjC^ is very small. Draw A^B^. We saw that

^-G^G^G win differ from 4-BGA by an infinitesimal (as B-^B

decreases) and ^GG^B^ will approach a right angle as a limit.

We thus get two approximate expressions for sinfl whosecomparison yields ^^ GG, '^^k^^'

for ^1— cos T BB^ is infinitesimal in comparison to BB^ or

* See figure on next page.

56 THE INTRODUCTION OF OH.

CCj. Again, we see that a line through the middle point,

of (AA^ perpendicular to AA^ will also be perpendicularto A^Cj, and the distance of the intersections will differ in-

finitesimally from sini/fil^j. We see that C^C^ differs by

a higher infinitesimal from sin^ cos rAA^, so that

cos T sin\If +(,= —^=- + e.

Fis. 2.

Next we see that AA^ = BB^, and hence

1bC0Sj-= . , cosy 4=2+e4.

* CO,

Moreover, by construction €^€2 = B^B^, CC^ = AA^. A per-pendicuW to AA^ from the middle point of (^^2) "^^ ^®perpendicular to A^B^, and the distance of the intersections

will differ infinitesimally from each of these expressions

siayjrAA^, £1^2

COS:


Hence 6 a eC08|- —cos r COST < e,

b a c ...COB T = cosT COB r • (»)

To get the Bpecial formula for the euclidean case, we shoulddevelop all cosines in power series, multiply through by P,

and then put p = 0, getting

b'^ = a'^ + c^

the usual Pythagorean formula.

We have now a sufficient basis for trigonometry, thedevelopment whereof merely requires a little analytic skill.

It may not perhaps be entirely a waste of time to work outsome of the fundamental formulae. Let A, B, C be thevertices of a triangle, and let us use these same letters, as

is usual in elementary work, to indicate the measures of the

corresponding angles, while the measures of the sides shall bea, b, c respectively. Begin by assuming that 4-ABG is a right

angle so that B = ^. Let D be such a point of {AC) that BDfit

is perpendicular to AG ; the measures of AB and GB being

6] and b^, while the measure of BB is a,.

a c

h '^^^l b °°^/fccos -r

= J cos -H = >

k a, k a,coB-r cos-r

k k

/bj + bns b a ccos ( I. ) = COSt = COStCOSt'

C0Sj^C0S^(l-COS^^) = ^COS^-^^ -COS'^^^COS^^^ -COB'^p

cos«^cos*j^(co8*-^ -2) = COB* j' -cos*^ -^Î'

(1-C08^|)(1-C0S«|C0S«|) = (1-COB^f) (l-COS^|),

. o, . 6 . a . csin-jr*sinr = sinTsmy

»

. a . a^

. b~ . csinj smr

58 THE INTRODUCTION OF oh.

Now proceeding with the AADB as we did with the A ABGwe shall reach two more sines whose ratio is

. a

and so forth. Continuing thus we have in (AB) and {AG)two infinite series of points. Let the reader show that the

limit for each series cannot be other than the point A itself.

Now we have just seen in (3) that the limit of this ratio

is sin^, hence

. a . b . . . .

sm r = sin r sin A. (5)

Let the reader deduce from (4) and (5) that

tan r = tan r cos .4. (6)

cos B = cos T sin A. {7)

Let us next suppose that AABG is any triangle. If noneof the angles be greater than a right angle, we may connect

any vertex with a point of the opposite side by a line

perpendicular to the line of that side, and we see at once that

a . b . c A • T) • /^sin r : siu r : sm r = 8in.4 : sin B : sm U.

k k k

Let us show that this formula holds universally, even when

this construction is not possible. Let us assume that B> — -

We may legitimately assume that A and C are less than ^

,

for the exti-eme case under the elliptic hypothesis where suchis not the fact may easily be treated after the simpler casehas been taken up. We shall still have

. a . c . . . ~sm T : sin r = sin A : sin C.

Let E be that point of (AG) which makes BE perpendicular

to AG. Let the measures of AE, BE, and GE be a', b', c',

while the measure of :ÂBE is A' and that of 4-GBEis C"

IT TRIGONOMETRIC FORMULAE 59

, b' ^ b'tan p "^î:

gobA'=J,

cosC' = -,tan Y tan j-

k k

.• a'. c'

8mil' = !l, 8mC =—^,• c . a

. b'

sin 5 = sin (4' + C") = ~^-^ ( COSIsin

I+ COS

ISin I )

,

c a' b' a c' b'cos r = COS -r cos J J cos T = cos -j- 008 j^ >

. „ k . fa c\

sin r sin -r

. , ,.6' .a.~ .c..

a' +c=o; sinT-= sm-rsmt/ = sin rsin^,

.a . b . csin 7 sin r sin ^

sin^ sin£ sinC(8)

Once more let us suppose that no angle of our triangle

is greater than a right angle, and let B be such a point of

{BG) that AD is perpendicular to BC:

COST

Ml>C c, cos —

=

cos 7"

6 k k

k~ vlBDcos ;

Ccosr'^^^k r a niBB . a . uBDl= cos

J-cos —

jh Sin T sin—r

—

M BB L K K K K Jcos —

i

a c , . a . c T,= cos -r cos r +Sm7;8inrC0SiJ.


li B > - this proof is invalid. Here, however, following

our previous notation

, ,6' a c . a' . c'

tan-'-r cos -r cos t —sin r- sin tT» /Ml rf-./v tC IC rC IC iC

COS B = co8{A' + C') = >

sinrsinr

a b' c' c V a' , . ,

COS T = cos r COS T ' cos-!- = cos J- COS T- J z= a +c

,

k k k k k k

. ,h' a' c' . a' . c'sin^ T-cos r- cos ^ —sin rsmr-

—

.

fC fC fC iC i€

cos i> =. a . csinjsin^

6 a cCOST — COStCOSt

"". o . c

^ a c. • a . c ^

cos |r = cos jr cos r + Bin TSin T cos B. (9)

A correlative formula may be deduced as follows :*

, . . a . h . cLet sin-T sm^ sin^

smA smB sin G

COB* r +\*sinM sin^CcoBî—ZX^sinil BinCcos£cos:r =k k

= COS'' -r coa^ T >

1 -A" sin25+ X*sin2^sin2C'co825-2X2Bin^8inCcos5cos t =k

= 1 -X^^ sin*^ -X2 sin^C-fX* sinfulsin^C,

sin*A + sin*C— sin*B

= sin*^ sin*C sin* ^ + 2sin^ sinC cosB cos j- ,

1-sinM -8in*C+ sin*^ sin*C

= sin*4sin*Cco8*7 — 28in4sinCcosrCos JB-fcos*£,

* I owe this ingenious trigonometric analysis to my former pupil Dr OttoDunkel.


cos4cob(7 = cos t sin -4. sin(7— cos £,

C085 = —cosjlcosC+sin^sinCcosT.* (10)

If ABGD be an isosceles birectangular quadrilateral, the

right angles being at A and B,

viCD kIO vlBD uAB . uAG. j,iBDcos —jT— = COS—r— COS —r— cos—r— + sm—r;— Sin —

v

(11)

The proof of this is left to the reader, as well as the task of

showing that the formulae which we have here established

are identical with those for a euclidean sphere of radius k.

Let him also show that when p = 0, our formulae pass over

into those for the euclidean plane.

* In finding this formula we have extracted a square root. To be surethat we have taken the right sign, we have but to consider the limitingcase .4 = 0, B = ir— C.

CHAPTER V

AJJALYTIC FORMULAE

At the beginning of Chapter I we posited the existence

of two undefined objects, points and distances. Between the

two existed the relation that the existence of two points

implied the existence of a single object, their distance. In

this relation the two points entered symmetrically.

These concepts may be further sharpened as follows.

Leaving aside the trivial case of the null distance, let us

imagine that a distinction is made between the two points,

the one being called the initial and the other the terminalpoint. The concept distance, where this distinction is madebetween the two points shall be called a directed distance,

or, more specifically, the directed distance from the initial

to the terminal point. Any not null distance will, thus,

determine two directed distances. The directed distance from

A to B shall be written AB. The relations congruent to

greater than, and less than, when applied to directed dis-

tances, shall mean that the corresponding distances have these

relations.

Suppose that we have two congruent segments (AB) and(A'B^ of the same line. It may be that a congruent trans-

formation which carries the line into itself, and transformsA and B into A' and B', also transforms A' into A. In this

case the middle point of (AA') will remain invariant, theextremities of every segment having this middle point willbe interchanged. Such a transformation shall be called areflection in this middle point. Conversely, we easily see

that a congruent transformation whereby A goes into A',and one other point of (AA') also goes into a point of thatsegment, is a reflection in the middle point of the segment.

There are, however, other congruent transformations of theline into itself besides reflections. For if A go into A', andany point of (AA') go into a point not of (AA'), then A willbe the only point of (AA') which goes into a point thereof,

there will be no invariant point on the line, and we havea difierent form of congruent transformation called a tranda-tion. It is at once evident that every congruent transformation

CH.V ANALYTIC FOEMULAE 63

of the line into itself is either a reflection or a translation.The inverse of a translation is another translation ; the inverseof a reflection is the reflection itself.

Theorem 1. The product of two translations is a translation.

The assemblage of all translations is a group.We see, to begin with, that every congruent transformation

has an inverse. This premised, suppose that we have atranslation whereby A goes into A', and a second wherebyA' goes into A". We wish to show that the product ofthese two is not a reflection. Suppose, in fact, that it were.A point Pj of (ÂA") close to A must then go into anotherpoint P3 of {AA") close to A". If A' be a point of {AA"), thefirst translation will carry P, into P^ a point of {A'A"), andas Pg is also a point of {A'A") the second transformationwould be a reflection, and not a translation. If A werea point of {A'A"), P^ would be a point of (AA'), and henceof {A'A"), leading to the same fallacy. If A" were a point of

(AA'), Pg would belong to the extension of (A'A") beyond A',

and Pg would belong to (A'A") and not to (AA").Let the reader show that the product of a reflection and

a translation is a reflection, and that the product of tworeflections is a translation.

Definition. Two congruent directed distances of the sameline shall be said to have the same sense, if the congruenttransformation which carries the initial and terminal points

of the one into the initial and terminal points of the other bea translation. They shall be said to have opposite ser^ses

if this transformation be a reflection. The following theoremis obvious

—

Theorem 2. The two directed distances determined by agiven distance have opposite senses.

Suppose, next, that we have two non-congruent directed

distances AB, A'C upon the same line, so that A'C > AB.There will then (XIII) be a single such point B of (A'C) that

AB = A'F. If then, AB and A'B' have the same sense, we

shall also say that AB and A'C' have the same sense, or

like senses. Otherwise, they shall be said to have opposite

senses. The group theorem for translations gives at once

—

Theorem 3. Two directed distances which have like or

opposite senses to a third, have like senses to one another,

and if two directed distances have like senses, a sense like

(opposite) to that of one is like (opposite) to that of the other,

64 ANALYTIC FORMULAE ch.

while if they have opposite senses, a sense like (opposite)

to that of one is opposite (like) to that of the other.

Let lis now make suitable conventions for the measurementof directed distances. We shall take for the absolute value

of the measure of a directed distance, the measure of the

corresponding distance. Opposite directed distances of the

same line shall have measures with opposite algebraic signs.

K, then, we assign the measure for a single directed distajice

of a line, that of every other directed distance thereof is

uniquely determined. K, further, we choose a fixed origin Dupon a line and a fixed unit for directed distances, everypoint P of the line will be completely determined by a single

coordinate —

>

. mOPX = sin

—

T—

•

In an entirely similar spirit we may enlarge our concepts ofangle, and dihedral angle, to directed angle. We choose aninitial and a terminal side or face, and define as rotations

a certain one parameter, group of congruent transformationwhich keep the vertex or edge invariant. We thus arrive

at the concept for sense of an angle, and Bet up a coordinatesystem for half-lines or half-planes of common bound. K in

the iÂBC,I

AB be taken as initial side, the resulting directed

ai^le shall be written lÂBG.We have at last elaborated all of the machinery necessary

to set up a coordinate system in the plane, and nearly all thatis necessary to set up coordinates in space. Let us begin withthe plane, and choose two half-lines

{OX,

\OY making a right

angle. Their lines shall naturally be called the coordinateaxes, while is the origin. Let P be any point of the plane,

the measure of OP being p, while those of 4-XOP aad ^YOPare a and /3 respectively. We may then put

f= fc sin -T cos a,

7 . P1/ = Asin^ cosjS, (1)

<^ = cos|,

with the further equation

P + 7l2 + /fc*«)« = A;2.

V ANALYTIC FORMULAE 65

In practice it is better to use in place of ^, ij, C homogeneouscoordinates defined as follows :—

-/x^^ + x^ + x^

v =kx2

-/x^^-^-XtI^-^-x^

What shall we say as to the signs to be attached to the

radicals appearing in these denominators ? In the hyperbolic

case &) is essentially positive, so that the radical must have thesame sign as x^. In the elliptic case it is not possible to havetwo points, one with the coordinates ^, t;, <t> and the other withthe coordinates —

f,— »j, —at, for their distance would be fcir,

and the opposite angle of every triangle containing them bothwould be straight, i.e. they might be connected by manysti-aight lines. On the other hand, it is not possible that

f, 7j, 0) and —f, — ?;, — o) should refer to the same point, for

then that point would determine with itself two distinct

distances, which is contrary to Axiom II. Hence, in every

case, the radical must have a well-defined sign in order that

equations should give a point of our space.

In the limiting parabolic case

f = p cos a, )j = p cos /3, a> = 1.

The formula for the distance of two points P and P' withcoordinates (a;), (a/) is

mPP' p p'. p . p , , .

cos—T— = cos Y cos T^ + sin— sm -^ cos (a — a)

= «M> + p

mPF a;o<+ai<+a!2< /,,cos i ^ •

,—

.—:• (O)

* '/x^^+x^+x} '/x^^+ x^'^+x.l''

. mPP'sm ^/

Xg ajj Xjj

^0 ^1 â

* -/xfvxf^ Vx^^+ x^^ + x^^'

(4)

The signs of the radicals in the denominators are, as wehave seen, well determined. The sign of the radical in the

numerator of (4), should be so taken as to give a positive


value to the whole. Should we seek the measures of directed

distances on the line PP', then, after the adjunction of the

value of the sign of a single directed distance, that of every

other is completely determined. In the euclidean case

•''0'''0

Returning to (4) and putting a"/= x^ + dzf we get for the

infinitesimal element of arc

t ^0 ^1 •''2

dt>^ _ I dxg dx-y dx^

Put x=~, y= —?j x'= z + dx> y'= y +dy

>

^ - , , iydx—xdyY

da^ =7, A. (5)

In the limiting euclidean case^ = 0,

ds^ = dx' + dy^.

Returning to the general case, we may improve our formula

(5) as follows :

—

let z=.v¥T^^T^. dz=<'^+y^y

.

If dx^ + dy^-dz^ = d,T\ d8 = ^^'^

z

Put u = = , V — J

—

- •

K—z k—zu' + v^ _ -2z4A« ~ k~z'

du^+ dif^= _^ [fje-zf \dx^ + dy"]

+ 2{k-z) {xdx-k-ydy)dz + {x^ + y'')dz^\

'-y-{dv?+dvf= [dx^^dy-^ j-^ - jj^^.dz^\

= d<f\

dv^+dv-'^d^^,.


ds' = [^+'^]'\d'^'+dv'). (6)

Comparing this with the usual distance formula

ds* = Edu^ + 2Fdudv + Gdv^,

Now if K be the measure of curvature of the surface havingthis distance formula

1 .JHogE SMog^.

LL 44* J

IT

Theorem 4. The non-euclidean plane may be developed upon

a surface of constant curvature r^ in euclidean space.

We shall return to questions of this sort in Chapters XVand XIX * of this work.

Let us now take up coordinates in three dimensions. Wemust make some preliminary remarks about the direction

cosines of a half-line. Suppose, in fact, that we have three

mutually perpendicular half-lines, \0X, \0Y, \0Z, and afom-th half-line \0P. The angles t-ÔP, 4.Y0P, 4.Z0Pwhose measures shall be a, /3, y respectively, shall be called

the direction angles of the half-line|OP. These angles are

not directed, but this will cause no inconvenience, as we shall

introduce them merely through the expressions cos a, cos/3,

cosy. These shall be called the direction cosines of the half-

line, shall be the origin, and OX, 07, OZ the coordinate

axes, while the planes determined by them are the coordinate

planes. Take a second half-line|OP', with direction cosines

cos a', cos /3', cos y'. We shall imagine that OP and OP' are

* The idea of interpreting the non-euclidean plane as a surface of constant

curvature in euclidean space must certainly have been present to Riemann's

mind, loc. cit. The credit for first setting the matter in u clear light is,

however, due to Beltrami. See his 'Teoria fondamentale degli spazii di

curvatura costante', Annali di Matematica, Serie 2, vol. ii, 1868, and 'Saggio

d'interpretazione della geometria non-euclidea ', Oiomale di MaUmaliche,

vol. vi, 1868.

£2


infinitesimaL Under these circumstances, tto may find

A, B, C where perpendiculars to the axes through P meetthem, and A', B", C bearing the same relation to P". Let Q' bethat point of

|OP' which makes 4-PQ'O a right angle, and let

4-POP' have a measure 0. Now we know that geometryin the infinitesimal domain obeys the euclidean hypothesis,

hence we have

mO^= hop coad + (,

the e is infinitesimal as compared with vlOP. In the same

^P""** MOg^= MaJcosa'+MOBcos^'+MOCcosy' + S.

But clearly mOA = MOP cos a + e, &c.

Hence

mOPcosO = MOP [cos a cosa' + cos ;3cos)3' + cosy cosy*] +7/,

or dividing out M OP,

cosfl = cosa cosa' + cos)3cos^+ cosycosy'. (7)

In particular we shall have

1 = cos* a + cos'^/S + cos'^y. (8)

We now set up our coordinate system as follows :

—

mOP0) = cos ; ,

k

, , . mOPf = K sm—-r— cos a,

r • ÔP ^7j = Asm

—J-— cosj3, (9)

. , . mOPC= « sm—7— cos y,

P = P+ ,*+C« + fcW.

From these we pass, as before, to homogeneous coordinatesa;^ : Xj : Xg : 0:3. But first we shall introduce a new symbol

:

{xy) = x^y^ + a;,2/,+ x^y^+ x^y^

.

(10)

We then write

a;-, kx^to = —pl=. , »j

=V(a.x) '/{xx)

V{xx) V{xx) ^ '

V ANALYTIC FORMULAE (B9

Here, as in the case of the plane, there is no ambiguity arising

from the double sign of the radical. There is, however, onemodification which we shall occasionally make. We see,

in fact, that in the hyperbolic case, since P < ; ^, ?;, C> <» a.re

real, we must have (asc) < 0, and Xg is a pure imaginary. Toremedy this let us write

KX(j ^ Xq, jBj ^ a!j, sBjj ^ ajg, x^ = x^.

A point will now have real coordinates. This distinction

between coordinates (x) and coordinates (x) shall be con-sistently maintained in the hyperbolic case.

The cosine of the measure of distance of two points (x) and

(y) is easily found. We see at once that we shall have

cos^=-J^l=. (12)* V{xx) V(yy)

Let us now see what effect a congruent transformation will

have upon our coordinates. First take a congruent trans-

formation keeping the origin invariant. We see at once that

the new direction cosines, and so the new coordinates (x'), will

bo linear functions of the old ones ; for a plane through the

origin will be characterized by a linear relation connecting

the direction cosines of the half-lines with that bound. Thevariables $, ri, C are thus linearly transformed in such a waythat $''+ r)^+C^ has a constant value, while a> is unaltered.

Hence x^, a;, , x^, x^ are linearly transformed so that (xx) is aninvariant (relative), i. e. they are subjected to an orthogonal

substitution.

Let us next suppose that we have a congruent transforma-

tion which carries the planes f = and ij = into themselves,

and every half-plane with this axis as bound into itself.

The assemblage of all such transformations will form a one-

parameter group, and this group may be represented by

d ^ . da> = <a cos 7 + fsin r >

v = »;,

»» , d ^ di = — a> sin r -f- fcos r

.

We see, in fact, that by this ti-ansformation every point

receives just the coordinates that it would obtain by a

translation of the axis OZ into itself through a distance d,

GO enlarged as to carry into itself every half-plane through

that axis. Once more wo find that, in the coordinates (a),

70 ANALYTIC FORMULAE CH.

this will be an orthogonal substitution. Now, lastly, every

congruent transformation of space may be compounded out

of transfoimations of these two types. Hence:

Theorem 5. Every congruent transformation of space is

represented by an orthogonal substitution in the homogeneousvariables ar„ : a^, : ajg : aij

.

In Chapter YIU we shall make a detailed study of these

congruent transformations. For the present, let us begin bynoticing that the coordinate planes have linear equations, andas we may pass from one of these to any other plane bylinear transformations, so the equation of any plane maybe written

(ux) =UgX^+ u^Xi + Wgajg + u^Xg = 0.

We see that (xy), (ux), (uv) are concomitants of everycongruent transformation, and we shall use them to find

expressions for the distance from a point to a plane and the

angle between two planes. The existence of the former of

these quantities is contingent upon the existence of a point

in the plane determining with the given point a line perpen-

dicular to the plane.

Let the plane (u) be that which connects the axis ajj = aig=with the point (y). Its equation is yzXi—y^x^Ô. Thecosines of the angles which this makes with the plane V|a;i=0are the x^ direction cosines of the two half-lines of OP. If

then, the measure of the angle be 0, we have

But both sides of this equation ai-e absolute invariants for all

congruent transformations. Hence, we may write, in general

:

(uv)coae= -^J=^-J-=- (13)

We find the distance from a point to a plane in the sameway. Let the point be (x) and d the distance thence to thepoint where a perpendicular to the plane u^x^^ = meets it,

this being, by de&iition, the distance from the point to theplane.

. d ^ x-i u^x,

K K V(xx) V{xx) v{uu)

Once more we have an invariant form, so that, in general

:

. d {uxSsin Y = — ' -^ ' (14)

« -/(Mit) V{xx) ^ '


The sign of "/{xx) is determined. As for that of -/{vm), by-

reversing it, we get opposite directed distances of the same line.

We have now reached the end of the first stage of ourjourney. Our system of axioms has given us a large bodyof elementary doctrine, a system of trigonometry, and asystem of analytic geometry wherein the fundamental metricalinvariants ai-e easily expressed. All of these things will be ofuse later. At present our task is different. We must showthat the system of axioms which has carried us safely so far,

will not break down later; i.e. that these axioms are essen-

tially compatible. We must also grapple with a disadvantagewhich has weighed heavily upon us from the start, renderingtrebly difficult many a proof and definition. Li Axiom XI weassumed that any segment might be extended beyond either

extremity. Yes, but how far may it be so extended ? This

question we have not attempted to answer, but have dealt

with the geometry of such a region as the inside of a sphere,

not including the surface. In fact, had we assumed that everysegment might be extended a given amount, we should haverun into a difficulty, for in elliptic space no distance may havea measure his under our axioms.

The matter may be otherwise stated. Every point will

have a set of coordinates in our system. What is the extremelimit of possibility for making points correspond to coordinate

sets, and what meaning shall we attach to coordinates to

which no point corresponds ? We must also adjoin the com-plex domain for coordinates, and give a new interpretation to

our fundamental formulae (12), (13), (14) covering the mostgeneral case. Then only shall we be able to continue our

subject in the broadest and most scientific spirit.

CHAPTER VI

CONSISTENCY A SIGNIFICANCE OF THE AXIOMS

The first fundamental question suggested at the close of

the last chapter -vras this. How shall we show that those

assumptions which we made at the outset are, in truth,

mutually consistent ? We need not here go into that elusive

question which bothers the modem student of pure logic,

namely, whether any set of assumptions can ever be shownto be consistent. All that we shall undertake to do is to

point to familiar sets of objects which do actually fulfil our

fundamental laws.

Let us begin with the geometry of the euclidean hypothesis,

and take as points any class of objects which may be put into

one to one correspondence with all triads of values of three

real independent variables x, y, z. By the distance of twopoints we shall mean the positive value of the expression

/{x'-xf + {i-yf+ {?^-zf.

The sum of two distances shall be defined in the arithmetical

sense. It is a perfectly straightforwai'd piece of algebra to

show that such a system of objects will obey all of our axiomsand the euclidean hypothesis; hence the consistency of ouraxioms rests upon the consistency of the number system,and that we may take as indubitable. Be it noticed that

we have another system of objects which obey all of ouraxioms if we make the further assumption that

a?-ity^-^z^<\.

The net result, so far, is this. If we take our fundamentalassumptions and the euclidean hypothesis, points and dis-

tances may be put into one to one correspondence withexpressions of the above types ; and, conversely, any systemof geometry corresponding to these formulae will be of the

euclidean type. The elementai-y geometry of Euclid fulfils

these conditions. In what immediately follows we shall

assume this geometry as known, and employ its teiTninology.

Let us now exhibit the existence of a system of geometryobeying the hyperbolic hypothesis. We shall take as our

CH, Ti AXIOMS 73

class of points the assemblage of all points in euclidean spacewhich lie within, but not upon, a sphere of radius unity.We shall mean by the distance of two points one half thereal logarithm of the numerically larger of the two cross ratios

which they make with the intersections of their line withthe sphere. The reader familiar with projective geometrywill see that the segment of two points in the non-euclideansense will be coextensive with their segment in the euclidean

sense, and the congruent group will be the group of collinea-

tions which carry this sphere into itself. Lastly, we see thatwe must be under the hyperbolic hypothesis, for a line is

infinitely long, yet there is an infinite number of lines througha given point, coplanaa* with a given Une, which yet do notmeet it.

The elliptic case is treated similarly. We take as points

the assemblage of all points within a euclidean sphere of

small radius, and as the distance of two points —. times, the

natural logarithm of a cross ratio which they determine withthe intersection of theii* line with the imaginary surface

sro*+V + a'2^ + «3'' = 0-

By a proper choice of the cross ratio and logarithm, this

expression may be made positive, as before. The congruentgroup will bo so much of the orthogonal group as carries

at least one point within our sphere into another such point.

The elliptic hypothesis will prevail, for two coplanar lines

perpendicular to a third will tend to approach one another.

We may obtain a simultaneous bird's-eye view of our three

systems in two dimensions as follows. Let us take for ourclass of points the assemblage of all points of a euclidean

sphere which are south of the equatorial circle. We shall

define the distance of two points in three successive different

ways :

—

(a) The distance of two points shall be defined as the

distance which the lines connecting them with the north pole

cut on the equatorial plane. A line will be a circle whichpasses through the north pole. If we interpret the equatorial

plane as the Gauss plane, we see that the congruent group

will be z'= 03 + ^, aa = 1,

or rather so much of this group as will carry at least one

point of the southern hemisphere into another such point,

ft is evident from the conformal nature of the transformation

from sphere to equatorial plane, that we are under the

euclidean hypothesis.

74 CONSISTENCY A SIGNIFICANCE ch.

(5) The distance of two points shall be defined as one half

the logarithm of the cross ratio on the circle through themin a vertical plane which they determine with the twointersections of this circle and the equator. A line here will

be the arc of such a circle. The congruent group will be

that gioup of (euclidean) collineations which carries into

itself the southern hemisphere. A line will be infinitely

long, yet there will be an infinite number of others through

any chosen point failing to meet it; i.e. we are underthe hyperbolic hypothesis.

(c) The distance of two points shall be defined as the length

of the arc of their great circle. Non-euclidean lines will bearcs of great circles. Congruent transformations will be

rotations of the sphere, and it is easy to see that the sumof the angles of a triangle is greater than a straight angle

;

we are under the elliptic hypothesis.

We have now shown that our system of axioms is sufBcient,

for we have been able to introduce coordinates for our points,

and analytic expressions for distances and angles. The axiomsare also compatible, for we have found actual systems of

objects obeying them. Compared with these virtues, all other

qualities of a system of axioms are of small import. It will,

however, throw considerable light upon the significance of

these our axioms, if we examine in part, their mutualindependence, by examining the nature of those geometricalsystems where first one, and then another of our assumptionsis supposed not to hold.

Axiom XIX is popularly known as the axiom of free

mobility, or rather, it is the residue of that axiom when weare confined to a limited space. It puts into precise shapethe statement that figures may be moved about freely withoutsuffering an alteration either in size or form. We have definedcongruent transformations by means of the relation congruentwhich is itself defined in the logical sense, but not de-scriptively. We might, of course, have proceeded in thereverse order.* The ordinary conception in the elementarytextbooks seems to be that two figures are congi'uent if theymay be superposed ; superposed means that they may becarried from place to place without losing size or shape, andthis in turn implies that throughout the transference, eachremains congruent to itself, fWith regard to the independence of this axiom, we have but

• Cf. Fieri, loc. cit.

t Cf. Veronese, loo. cit., p. 259, note 1, and Russell, The Principles o/Mathe-maKes, toI. i, Cambridge, 1908, p. 405.

VI OF THE AXIOMS 75

to look at any system -where the measure of distance in onepiano is double that of all the rest of space. A triangle havingtwo vertices in this plane, and one elsewhere, could not becongruently transformed into a triangle of a different sort.

Axiom XVin is the axiom of continuity. We have laid

special stress on it in the course of our work, although thesubject of elementary geometry may be pushed very far

without its aid.* We are not here concerned with thequestion of the wisdom of such attempts, considered fromthe didactic point of view. Systems of geometry where this

axiom does not hold will occur to every reader; e.g. the

Cartesian euclidean system where all points whose coordinates

are non-algebraic are omitted. It is interesting to note that

whereas the omission of XIX runs directly counter to oursense experience, no amount of observation could tell us

whether or no our geometry were continuous, fAxiom XVII is an existence theorem, not holding where

the geometry of the plane is alone considered. It is a verycurious fact that the projective geometry of the plane is notentirely independent of that of space, for Desargues' theoremthat copolar triangles are also coaxal cannot be provedwithout the aid either of a third dimension, or of the con-

gruent group.JAxiom XVI gives a criterion for circumstances under which

two lines must necessaiily intersect. It is evident that

without some such criterion we should have difficulty in

proceeding any distance at all among the descriptive pro-

perties of a plane. It is difficult to show the independence

of this axiom. The only dense system of geometry knownto the writer where it is untrue is the following. §

Let us denote by R the class of all rational numbers whosedenominators are of the form

where a^ and b^ are integers or one may be zero. Let us

take as points the assemblage of all points of the euclidean

plane whose Cartesian coordinates are rational numbersof the class R. The whole field will be transported into

itself by a parallel translation from any one point to anyother. Moreover, let x, y and x', ^ be the coordinates of two

* Cf. Halsted, loc. cit

f Cf. B. L. Moore, loc. cit.

t Cf. Hilbert, loc. cit., p. 70 ; Moulton, ' A simple non-desarguesian plane

geometry,' Tranaactiota of the American Uathemaiical Society, vol. iii, 1902;

VahJen, loc. cit., p. 67.

i Cf. Levy, loc. cit., p. 32.

76 CONSISTENCY AND SIGNIFICANCE ch. vi

points of the class, where x^ + y' = x'* + y'^. We may imagine

in fact that

x = -, y = -. X--, y--, --^_-—^.Then the cosine and sine of the angle which the two points

suhtend at the origin will be respectively

pp'+gq' pq'-p'q

p^ + q" ' p'+ q"'

and these are numbers of the class R. The whole field will

go into itself by a rotation about the origin. Our system

will, therefore, obey XIX. It is of course two-dimensional

and not continuous. Moreover XVI will not hold, as the

reader will see by easily devised numerical experiments.

There are, also, plenty of geometries of a finite numberof points where this axiom does not hold.*

Axiom XV is, of course, an existence theorem, untrue in the

geometry of a single line.

Axiom XIV gives the fundamental property of straight

lines. As an example of a geometry where it does not hold,

let UB consider the assemblage of all points within a sphere

of radius one, and define as the distance of two points the

length of an arc of a circle of radius two which connects them.The segment of two points is thus a cigar-shaped region

connecting them. We see that the extensions of such a seg-

ment and the segment itself do not comprise the segmentof two points within the original, and the extensions of the

latter. Axioms XII and XIII are also in abeyance, and it

seems possible that these three axioms are not mutuallyindependent. The present writer is unable to answer this

question.

Axiom XI implies that space has no boundary, and wUl beuntrue of the geometry within and on a sphere.

The first ten axioms amount to saying that distances aremagnitudes among which subtraction is always possible, butaddition only under restriction.

* Veblen, loc. cit, pp. 860-51.

CHAPTER VII

THE GEOMETRIC AND ANALYTIC EXTENSIONOF SPACE

Wk are now in a position to take up the second of thosefundamental questions which we proposed at the close of

Chapter V, namely, to determine what degree of precision

may be given to Axiom XI. This axiom tells us that,

popularly speaking, any segment may be extended beyondeither end. How far may it be so extended? Are we able

to state that there exists a system of geometry, consistent

with our axioms, where any segment may be extended by

any chosen amount? Or, in more precise language, ii ABand PQ be given, can we always find C so that

AG=AB+BG, BC = PQ.

We are already able to answer this question in the euclidean

case, and answer it aflSrmatively. We have seen that there

is no inconsistency in that system of geometry, where points

are in one to one con-espondence with all triads of (real andfinite) values of three coordinates x, y, z, and where distances

are given by the positive values of expressions of the form

^/{x'-xf + {y'-yf+ (z'-zf.

Here, if, as we have said, we restrict the values of x, y, z

merely to be real and finite, we have a space under the

euclidean hypothesis, where any segment may be extended

beyond either extremity by any desired amount. Such aspace shall be called euclidean space.

The same result will hold in the hyperbolic case. We shall

have a consistent geometrical system if we assume that our

points are in one to one correspondence with values

x^:Xi:x.^:x^, h^<0,

h^Xo^+x^+x^ + x^^<0.

Here, also, there will exist on every line distances whosemeasures will be as large as we please. The space under the

78 THE GEOMETRIC AND ANALYTIC ch.

hyperbolic hypothesis, where any segment may be extended

by any chosen amount shall be called hyperbolic space. Toput the matter otherwise, we shall have euclidean or hyper-

bolic geometry if we replace Axiom XIT by :

—

Axiom XII'. If the parabolic or hyperbolic hypothesis be

true, and if AB and FQ be any two distances, then there

will exist a single point C, such that

AO = AB+BC, BG = FQ.

When we turn to the elliptic case, we find a decidedly

different state of affaii-s. Suppose, in fact, that there is a one

to one correspondence between the assemblage of all points,

and all sets of real values x^-.xîx^-.x^. The distance of twopoints will depend upon the periodic function

cos-^-—M=.V{xx) v{yy)

If, to avoid ambiguity, we assume that the minimum positive

value should be taken for this expression, we should easily

find two not null distances, whose sum was a null distance,

which would be in disagreement with Axiom X.The desideratum is this. To find a system of geometry

where each point belongs to a sub-class subject toAxioms I-XIX,and the elliptic hypothesis, and where each segment may still

be extended by any chosen amount, beyond either end.

Axiou I. There exists a class of objects, containing at

least two members, called points.

Axiom II'. Every point belongs to a sub-class obeyingAxioms I-XIX.

Definition. Any such sub-cla^B shall be called a consistent

region.

Axiom III'. Any two consistent regions which have acommon point, have a common consistent region includingthis point and all others determining therewith a snfflciently

small, not null, distance.

Axiom IV'. if P„ and P„+i be any two points there maybe found a finite number n of points P^,P^,P^...P^ possessingthe property that each set of three successive ones belong toa consistent region, and P^ is within the segment (Pfc_i -P^+i)-

Definition. The assemblage of all points of such segments,and all possible successive extensions thereof shall be called

a line.

VII EXTENSION OF SPACE 79

An impoi'tant implication of the last axiom is that any twopoints may be connected (conceivably in many ways) bya chain of consistent regions, where each successive pairhave a consistent sub-region in common. This shows thatif we set up a coordinate system like that of Chapter Y in

any consistent region, we may, by a process of analytic ex-

tension, reach a set of coordinates for every point in space.

We may also compare any two distances. We have merelyto take as unit of measure for one, a distance so small, that

a distance congruent therewith shall exist in the first three

overlapping consistent regions; a distance congruent withthis in the second three and so on to the last region, and thencompare the measures of the two distances in terms of the

fi.rst unit of measure, and the unit obtained from this by the

series of congruent transformations. Let the reader show that

>if once we find AB = FQ the same relation will hold if we

<proceed by any other string of overlapping regions. Havingthus defined the congi-uence of any two distances, we maystate our axiom for the extension of a segment, as follows :

—

Axiom V. If AB and PQ be any two distances, there

exists a single point C such that BO = PQ, while B is

within a segment whose extremities are C and a pointot{AB).An important corollary from this axiom is that there

must exist in the elliptic case a point having any chosen set

of homogeneous coordinates (a;) not all zero. For, let {y) bethe coordinates of any known point. Consider the line

through it whose points have coordinates of the form

\(y)+li{x). As we proceed along this line, the ratio - will

always change in the same sense, for such will be the case

in any particular consistent region. Moreover we may, byour last axiom, find a number of successive points such that

the sum of the measures of their distances shall be k-n.

Between the first and last of these points the value of - will

have run continuously through all values from — ao to oo

,

and hence have passed through the value 0, giving a point

with the required coordinates.

The preceding paragraph suggests two interesting questions.

Is it possible that, by varying the method of analytic ex-

tension, we might give to any point two different sets of


homogeneous coordinates in the same system 1 Is it possible

that two dififerent points should have the same homogeneous

coordinates? With regard to the first of these questions, it

is a fact that under our hypotheses a point may have several

difierent sets of coordinates, as we shall see at more length

in Chapter XVII. For the present it is, however, wiser to limit

ourselves to the classical non-euclidean systems, where a point

has a unique set of coordinates. We reach the desired

limitation by means of the following considerations.

A sufficiently small congruent transformation of any con-

sistent region will effect a congruent transformation of any

chosen sub-region, and so of any consistent region including

this latter. It thus appears that if two consistent regions

have a common sub-region, a sufficiently small congruent

transformation of the one may be enlarged to be a congruent

transformation of the other. Proceeding thus, if we take anytwo consistent regions of space, and connect them by a series

of overlapping consistent regions, then a small congruent

transformation of the one may be analytically extended to

operate a congruent transformation in the other. Will the

original transformation give rise to the same transformation

in the second space, if the connexion be made by means of

a different succession of overlapping consistent regions ? It

is impossible to answer this question a priori ; we therefore

make the following explicit assumption:

—

Axiom VI'. A congruent transformation of any consistent

region may be enlarged in a single way to be a congruenttransformation of every point.

Evidently, as a result of this, a congruent transformation

of one consistent region can be enlarged in only one wayto be a congruent transformation of any other. Let us nextobserve that it is impossible that two points of the sameconsistent region should have the same coordinates in anysystem. Suppose, on the contrary, that P and Q of a con-sistent region have the coordinates (a;). There will be nolimitation involved in assuming that the coordinate axes wereset up in this consistent region, and the coordinates of P founddirectlyas inChapterV,while those ofQ are found byan analyticextension through a chain of overlapping consistent regions.

Now it is not possible that every infinitesimal congruenttransformation which keeps P invariant shall also keep Qinvariant, so that a transformation of this sort may be foundtransforming each overlapping consistent region infinitesimally,

and carrying Q to an infinitesimally near point Q'. But in


the analytic expression of this transformation, in the formof an orthogonal substitution (in the non-euclidean cases)

the values (a:) will be invariant, so that Q" will also havethe coordinates (o;), and by the same chain of extensions as

gave these coordinates to Q. Hence, reversing the order of

extensions, when we set up a coordinate system in the last

consistent region, that which includes Q and Q', these twopoints will have the same coordinates. But this is impossiblefor the coordinate system explained in Chapter V, for a con-sistent region gives distinct coordinates to distinct points.

This proof is independent of Axiom VI'.

Our desired uniqueness of coordinate sets will follow at

once from the foregoing. For, suppose that a point P havetwo sets of coordinate values (a;) and (a;'), not proportional

to one another. Every infinitesimal transformation whichkeeps the values (a;) invariant, wiU either keep (a;') invariant,

or transform them infinitesimally, let us say, to a set of

values (a;"). But there is a point distinct from P and close to

it which has the coordinates (a;"), and this gives two points

of a consistent region with these coordinates, which we havejust seen to be impossible. Hence, the ratios of the coordinates

(aô') must be unaltered by every infinitesimal orthogonal

substitution which leaves (a;) invariant, i.e. a*o'= px^. It is

evident, conversely, that if each point have but one set of

coordinates. Axiom VI' must surely hold.

It is time to attack the other question proposed above,

by supposing that two distinct points shall have the samehomogeneous coordinates. They may not lie in the sameconsistent region, and every congruent transformation whichleaves one invariant, will leave the other unmoved also.

Let us call two such points equivalent. Eveiy line through

one of these points will pass through the other. For let

a point Q on a line through one of the points have coor-

dinates {y). We may connect it with the other by a line,

and the two lines through (Q) lie in part in a consistent

region, the coordinates of points on each being represented

in the form Aj/i + Mî' The two lines are identical.

Let us consider the assemblage of all points whose coor-

dinates are linearly dependent on those of three non-collinear

points. This assemblage of points may properly be called

a jAane, for those points thereof which lie in any consistent

region will lie in a plane as defined in Chapter II. It is

clearly a connex assemblage, and will contain every line

whereof it contains two non-equivalent points. Let {y), [z), {t)

be the coordinates of three points, no two of which are

82 THE GEOMETRIC AND ANALYTIC CH.

equivalent. Let us consider the point (x) whose coordinates

*'"®(ux) =

I

uyztI

.

In the elliptic case, as we have seen, such a point surely

exists. In the hyperbolic or parabolic cases, there might not

be any such point. It is clear, however, that in these cases,

there can be no equivalent points. Suppose, in fact, Pg andP^^. were equivalent. Connect them by a line whereon are

Pj, P2...P„. Move this line slightly so that the connecting

string of points are P^,P^ ...P^, very near to the former

points. We have constructed two triangles, and {n— \)

quadrilaterals, and as we are under the hyperbolic or euclidean

hypothesis, the sum of the measures of the angles of all the

triangles and quadrilaterals will be less than, or equal to

Tr+ (7j,— l)2ir + ir. But clearly the sum of the measures of

the angles at points Pj and Pj is 27). ir, so that the sum of the

two angles which the two lines make at P(, and P„+i is null

or negative ; an absurd result. Equivalent points can thenoccur only under the elliptic hypothesis, and there will surely

be a point P with the coordinates {x) above.

Let us next make a congruent transformation whereby Pgoes into an equivalent point P', the plane of (i/) (z) (f) goesinto itself congruently, for it constitutes the assemblage of all

points satisfying the condition {xX) = 0, and (xA) is aninvariant under every orthogonal substitution. After P hasbeen carried to P', each point of the plane may be returnedto its original position by means of a series of congruenttransformations, each too small to change P' to an equivalentpoint, yet keeping the values {x) invariant, coupled, at theend, with a reflection in a plane perpendicular to the givenone, in case the determinant of the original orthogonalsubstitution is negative, and this too will leave P' unchanged.We may therefore pass from P to any equivalent point bya transformation which leaves in place every point of a plane.

But there is only one congruent transformation of spacewhich leaves every point of a plane invariant, besides, ofcourse, the identical one. Hence every point in space canhave but one equivalent at most.

Our results are, then, as follows. Under the euclidean andhyperbolic hypotheses, there is but one point for each set

of coordinates, and our new Axioms I-VI' will yield usnothing more than euclidean or hyperbolic space. Under theelliptic hypothesis there are two possibilities :

—

Elliptic gpcLce. This is a space obeying Axioms I-VI', andthe elliptic hypothesis. If n successive segments whose


kitmeasures are— be taken upon a line as indicated in V, the

n '^

last extremity of the last segment will be identical withthe first extremity of the first. Two lines of the same plane

will have one and only one common point, so that no point

has an equivalent. We may take as a consistent region the

assemblage of all points whose distances from a given point

kirare of measure less than —r- . If two points be of such a

4nature that the expression for the cosine of the measure of

the kth part of their distance vanishes, we shall say that the

measure of their distance is -^. Two points will alwaysi&

have a determinate distance and a single segment, unless the

Kirmeasure of their distance is -5- , in which case they determine

two segments with the same extremities. These last twosegments may also, with propriety, be called half-lines. Thedefinition of an interior angle given in Chapter II may beretained, but the concept of halt-plane is iUusory, for a line

will not divide the plane. It may, however, be modified

much as we have modified the definition of a half-line, andfrom it a definition built up for a dihedral angle. We leave

the details to the reader. An example of elliptic geometrywill be furnished by any set of points in one to one corre-

spondence with all sets of homogeneous values x^-.x-^xx^: x^

where also cos r = — — - For instance, let us take as« -/(xx) -/(yy)

points concurrent lines of a four dimensional space (euclidean,

for example) and mean by distance the measure of the angleIT

^ ^ formed by two lines.

Spherical space. This is also a space obeying Aôms I-VI'and the elliptic hypothesis. Each point wiU have one equiva-

lent. If n successive congruent distances be taken upon

a line whose measures are— . the last extremity of the last

will be equivalent to the first extremity of the first. Wemay take as a consistent region the assemblage of all points

the measures of whose distances from a given point are less

/fcwthan -^ . The measure of the distance of two equivalent

points shall be defined as the number kv. Any two non-f2


equivalent points will have a well-defined segment. We mayfind a definition for a half-line analogous to that given in

the elliptic case, and so for half-plane, internal angle, anddihedi'al angle.

An example of spherical geometry will be furnished by the

geometry of a hypersphere in four dimensional euclidean

space, meaning by the distance of two points, the length

of the shorter arc of a great cii-cle connecting them.

A simple example of a two dimensional elliptic geometryis offered by the euclidean hemisphere, where opposite points

of the limiting great circle are considered as identical. A twodimensional spherical geometry is clearly offered by the

euclidean sphere.

The elliptic and spherical spaces which we have thus built

up are, in one respect, more complete than euclidean or

hyperbolic space, in that there is in the first two cases alwaysa point to correspond with every set of real values, not all

zero, that may be attached to our four homogeneous coor-

dinates X, while in the latter cases this is not so. We bring

our euclidean and hyperbolic geometries up to an equality

with the others by extending our concept point. Let us beginwith 'the euclidean case where there is a point corresponding

to every real set of homogeneous values xîXi-.x^'-x^, pro-

vided that Xg ^ 0. Now a set of values O-.y^-.y^-.y^ will

determine at each real point (x) a line, the coordinates of

whose points are of the form XyQ+nXf, and if (x) be varied

off of this line, we get a second line coplanar with the first.

Our coordinates Oiy^-.yîys will thus serve to determine

a bundle of lines, and this will have exactly the samedescriptive properties as a bundle of concurrent lines. Wema}' therefore call the bundle an ideal point, and assign to

it the coordinates (y). Two ideal points will determine apencil of planes having the same descriptive properties as

a pencil of j)lanes through a common line. We shall there-

fore say that they determine, or have in common, an ideal

line. Two lines whose intersection is ideal shall be said

to be parallel, as also, two planes which meet in an ideal

line. These definitions of parallel are for euclidean spaceonly. The assemblage of all ideal points will be characterized

by the equation _ f.Xq— u.

This we shall call the equation of the idâl plane which is

supposed to consist of the assemblage of all ideal points.

Ideal points and lines shall also be called infinitely distant,

while the ideal plane is called the plane at infimly. We shall

vii EXTENSION OF SPACE 85

in future use the words point, line, and jjiane to cover bothideal elements and those previously defined, which latter maybe called, in distinction, actual. Actual and ideal elementsstand on exactly the same footing with regard to purelydescriptive properties. No congruent transformation caninterchange actual and ideal elements. We shall later return

to the meaning of such words as distance where ideal elementsenter.

In the hyperbolic case we may apply the same principles

with slight modification. There will be a real point corre-

sponding to each set of real homogeneous coordinates {x) for

which ji.ij.2 +ii + î + igZ < 0.

A set of real homogeneous values for {x), for which this

inequality does not hold, will determine a bundle of lines,

one through every actual point, any two of which are

coplanar; a bundle with the same descriptive properties as

a bundle of concurrent lines. We shall therefore say that

this bundle determines an ideal point having the coordinates

the ideal point shall be said to be infinitely distant. If

the ideal point shall be said to be ultra-infinite. Two lines

having an infinitely distant point in common shall be called

parallel. Through each actual point will pass two lines

parallel to a given line. An equation of the type

(iLx) = 0, iV +< + Ws' + %' > 0=

will give a plane. If the inequality be not fulfilled, the assem-blage of all ideal points whose coordinates fulfil the equation(and there can be no actual points which meet the requirement)

shall be called an ideal plane, the coefiicients (u) being its

coordinates. There will thus be a plane corresponding to

each set of real homogeneous coordinates (u) not all zero.

An ideal line shall be defined as in the euclidean case, andthe distinction between actual and ideal shall be the sameas there given. No congruent transformation, as defined so

far, can interchange actual and ideal elements.

Let us take account of stock. By the introduction of ideal

elements we have made, each of our spaces a real analytic

continuum. In all but the spherical case there is a one to

one correspondence between points and sets of real homo-geneous values not all zero, in spherical space there is a one


to one correspondence of coordinate set and pair of equivalent

points. Each of our spaces vdll fulfil the fundamental

postulates of projective geometry, as we shall develop themin Chapter XVIII, or as they have already been developed

elsewhere.* Let us show hurriedly, how to find figures to

coiTespond to imaginary coordinate values. Four distinct

points will determine six numbers called their cross ratios,

which have a geometrical significance quite apart from all

concepts of distance or measurement.f An involution will

arise when the points of a line are paired in such a reciprocal

manner that the cross ratios of any four are equal to the

corresponding cross ratios of their four mates. If there be

no self-corresponding points, the involution is said to be

elliptic. If the points of a line be located by means of

homogeneous coordinates A : Mi it^ T^^y be shown that everyinvolution may be expressed in the form

In particular if (y) and (z) be the coordinates of two points,

there will exist an involution on their line determined by the

equations(^^) ^ ^y) ^ ^,^^)^ (xy= y.iy)-X{z),

and by a proper choice of running coordinates any elliptic

involution may be put into this form. Did we seek the

coordinates of self-corresponding points in this involution,

we should get{x) = {y)±i{z).

Conversely, every set of homogeneous complex values {y) + i{z)

will lead us in this way to a definite elliptic involution.

The involution may be taken to represent the two sets ofconjugate imaginary homogeneous values. We may separate

the conjugate values by the following device. It is not difficult

to show that if a du'ected distance be determined by twopoints, it will have the same sense as the correspondingdirected distance determined by their mates in an elliptic

involution. To an elliptic involution may thus be assignedeither one of two senses of description, and we shall define

as an imaginary point an elliptic involution to which sucha sense has been attached. Had we taken the other sense,

we should have said that we had the conjugate imaginary

* Cf. Fieri, 'I prinnipi della geometria di posizione.' Memorie deUaR. Accademia deUe Scieme di Torino, yol. xlviii, 1899.

t Cf. Pasch, loc. cit., p. 164, and Chapter XVIII of the present work.The idea of assigning to four collinear points a projectively invariantnumber originated with Von Staudt, Beitr&ge zur Oeomdrie dtr Lage, Fart 2,§§ 19-22, Erlangen, 1868-66.

vn EXTENSION OF SPACE 87

point. An imaginary plane may similarly be defined as anelliptic involution among the planes of a pencil, with aparticular sense of description; an imaginary line as theintersection of two imaginary planes. It may be showngeometrically that by introducing imaginary elements underthese definitions we have a system of points, lines, and planes,

obeying the same descriptive laws of combination as do thereal points of lines and planes of projective geometry, orthe assemblage of all real homogeneous coordinate sets, whichdo not vanish simultaneously.* Introducing these imaginaryexpressions, and the corresponding complex values for their

homogeneous coordinates, we extend our space to be a perfect

analytic continuum.We must now see what extension must be given to the

concept distance, in order to fit the extended space withwhich we are, hencefoi-th, to deal. To begin with, we shall

from this time forth identify the two concepts distarice andvneasare of distance. In other words, as the concept distance

comes into our work efiectively only in terms of its measure,

i e. as a number, so we shall save circumlocution by replacing

the words measure of distance by distance throughout. Thedistance of two points is thus dependent upon the two points,

and on the unit. In any particular investigation, however,we assume that the unit is well known from the start, anddisregai-d its existence. We therefore give as the definition

of the distance of two points under the euclidean hypothesis

«« = rv A^h-y^'+i^z-y^'+i^-vsr- (i)

•''OilO

This is, at worst, a two valued function. When it takes

a real value, we give the positive root as the distance, whenit is imaginary we may make any one of several simpleconventions as to which root to take. If one or both of the

points considered be ideal, the expression for distance becomesinfinite, unless also the radical vanishes when no distance is

determined. Under these circumstances we shall leave the

concept of distance undefined, thus getting pairs of points

disobeying Axiom II'. Notice also that whenever the radical

vanishes for non-ideal points we have points which are

distinct, yet have a null distance, and when such points

are included. Axiom XIII may fail.

We shall in like manner identify the concepts angle and

* Cf. Von Staudt, loc. cit. , S 7, and Luroth, ' Das Imagin&re in der Geometriennd das Beclmen mit Wurfen,' MaUumatiache Annalen, Yol. is.

88 THE GEOMETRIC AND ANALYTIC CH.

measure of angle in terms of the unit which gives to a right

angle the measure - •

We may proceed in a similar manner in the non-euclidean

cases. If (x) and (y) be the coordinates of two points, weshall define as their distance c^, the solution of

cos Y = • ("«/

« V{xx) V(yy)

This equation in d has, of course, an infinite number of

solutions. Before taking up the question of which shall be

called the distance of the two points, let us approach the

matter in a different, and highly interesting fashion due to

Cayley.* This theory is of absolutely fundamental impor-

tance in all that follows.

The assemblage of points whose coordinates satisfy the

«1"^*^°°(XX) = 0, (3)

shall be called the Absolute. This is a quadric surface, real

in the hyperbolic case, surrounding, so to speak, the actual

domain ; imaginary in the elliptic and spherical cases ; in the

last-named, it is the locus of points which coincide with their

equivalents. Every congruent transformation is an orthogonal

substitution, i.e. a linear transformation can-yingthe Absolute

into itself. Let us, by definition, enlarge our congruent groupso that every such transformation shall be called congruent

;

certainly it carries a point into a point, and leaves distances

unaltered. In the euclidean case we take as Absolute the

<'°'^<'x, = Q, x^^ + x^^ + xi = Q, (4)

and define as congruent transformations a certain six-parametersub-group of the seven-parameter coUineation group whichcanies it into itself. We shall return to the study of the

congruent group in the next chapter.

Betuming to the non-euclidean cases, let us take twopoints Pj, ?2 'wîtli coordinates {x) and (y), and let the line

connecting them meet the Absolute in two points Qi,Q2- Weobtain the coordinates of these by putting \{x) + ix{y) into

the equation of the Absolute. The ratio of the roots of this

equation will give one of the two cross ratios formed bythe pair of points P^P^ and the pair Q^Q^ ; interchanging

* Cayley, ' A sixth memoir on Quantics;,' Philosophical Transadions of theRoyal Society of London, 1859.

vii EXTENSION OF SPACE 89

the roots we get the other cross ratio of the two pairs ofpoints *. The value of such a cross ratio will thus be

(ocy)- V{xyf-{xx) (yy)

By interchanging the signs of the radicals we change this

cross ratio into its reciprocal, and this amounts to inter-

changing the members of one of the two point pairs. Let us

denote this expression by e fc •

cf ^ (^y) + Axx)(yy)-(xyf^

V{xx) V{yy)

cos^= ,-^\— . (5)« V{xx) V(yy)

If we write the cross ratios of the pair of points Pj P^ andthe pair QjQ^ ^^ {^1^2' QiQz)) '^^ ™*y re-define our non-euclidean distance by the following theorem :

—

TheoreTn. If d be the distance of two points Pj and P,whose line meets the Absolute in Q^ and Qj,

d = ^log,{P,P„QM. (6)

The great beauty of this definition is that it brings into

clear relief the connexion between distance and the congruentgroup, for the cross ratio in question is, of course, invariant

under all linear transformation which carry the Absoluteinto itself, i.e. under all congruent transformations. Let the

reader show that a corresponding projective definition maybe given for an angle.

Our distances, as so far defined, are infinitely multiple

valued functions. There is no great practical utility in

rendering them single valued by definition. It is, however,perhaps worth while to carry it through in one case.

If we have two real points of the actual domain, the

expression {PiPzi Q1Q2) 'wiH have two values, real in the

hyperbolic, pure imaginary in the elliptic and spherical case,

and these two are reciprocals, so that the resulting expressions

for d will diflfer only in sign, for each determination of the

logarithm. We may therefore take the distance as positive.

* For the geometrical interpretation of a crosa ratio when some of the

elements are imaginary, see Von Staudt, loc. cit., $ 28, and Liiroth, loc. cit.


Did we seek, not for a distance, but a directed distance, then

it would be necessaiy to distinguish once for all between

Q, and Q^ and in each particular case between the pair Pi-Pj,

and the pair PiP-i^ the directed distance will have a definite

value sometimes positive, sometimes negative.

Let us specialize by confining ourselves to the hyperbolic

case. We have defined the distance of two actual points.

Still restricting ourselves to the real domain, suppose that

we have an actual and an ultra-infinite point. Let us choose

such a unit of measure that k^ = — 1. Our cross ratio is here

negative, with an absolute value r let us say, so that the distance

expression takes the form ^ [logr + (2m+l)iri]. Letus choose

in particular

d = ^logr+ — •

Next consider two ultra-infinite points. If the line con-

necting them meet the Absolute in real points, we shall havea real cross ratio as before, and hence a real positive distance.

If, however, this real line meet the Absolute in conjugateimaginary points, the expression for the cross ratio becomesimaginary, and the simplest expression for their distance is

pure imaginary. The absolute value of this expression will

run between and — » for the roots of ^ logA =z X differ

by -ni. We may, hence, represent all of these cross ratios in

the Gauss plane by points of the axis of pure imaginaries

between and —-•

If the line connecting two ultra-infinite points be tangentto the Absolute, the cross ratio is unity, and we may takethe distance as zero. The distance from a point of theAbsolute to a point not on its tangent will be infinite;

the distance to a point on the tangent is absolutely inde-terminate, for the cross ratio is indeterminate. We may,in fact, consider the cross ratios of three coincident pointsand a fourth, as the limiting case of any cross ratio whichwe please.

Leaving aside the indeterminate case, we are thus able to

represent the distance of any two real points of hyperbolicspace in the Gauss plane by a point on the positive halfof the axis of reals, by a point of the segment of the origin

and - i, or by a point of the horizontal half-line — i ck>,


and as two points move continuously in the real domain of thehyperbolic plane, the points -which represent their distancewill move continuously on the lines described.

Let us now take two points of the hyperbolic plane, real orimaginary. We see that the roots of ^logJ. = Z differ bymultiples of vi, so that we may assign to d an imaginary

part whose Absolute value ^ - Moreover, by choosing

properly between the two reciprocal values of the cross ratio,

we may ensure that the real part of d shall not be negative.

If two points be conjugate imaginaries, while their line cuts

the Absolute in real points, the cross ratio is imaginary, andthe expression for distance is pure imaginary, which we mayrepresent by a point of the segment of the origin and

IT— - i. If both pairs of points be conjugate imaginaries, the

cross ratio is real and negative, so that the distance may

be represented in the form X— -î. We shall define as the

distance of two points that value of the logarithm of a cross

ratio which they form with the intersection of their line andthe Absolute, which in the Gauss plane is represented by

a point of the infinite triangle whose vertices are oo , + ^ i >

w .... ^— —i. The possible ambiguities for points on the sides of

this triangle have already been removed by definition.

We have already seen that when euclidean space has beenenlarged to be a perfect analytic continuum, imaginary points

and distances come in which do not obey all of our axioms.

In the hyperbolic case we shall find real, though ultra-infinite,

points which do not at all obey the principles laid downfor a consistent region.* Let us take three points of the

ultra-infinite region of the actual hyperbolic plane x^ = 0,

say (x), (y), (z). As these points are supposed to be real wemay assume that x^, x^ are real, while x^ is a pure imaginary,

and that a like state of affairs exists for {y) and (z). Weshall farther assume that the lines connecting them shall

intersect the Absolute in real, distinct points. We have then

{yzf- (yy) (^2) > 0. (««) > 0,

(sxf-{zz)(xx)>0, (yy)>0, (7)

{xyY-{xx){yy) > 0, (zz) > 0.

* The developments which follow are taken from Study, 'Beitrftge zur

nicht-euUidischen Oeometrie,' American JounuU ofMtUhemalics, vol. zziz, 1907.


Let us, for the moment, indicate the distance from (x) to {y)

by 03^, and assume yz^sx^^.We shall also take

h = i, cos 7 = cosh d.K

Under what circumstances shall we have ?

yz ^zx + xy,

cosh (yz—zx) ^ cosha^,

/_ML /ISl _ IZSZV iyy){zz) V (2z){xx) V {xx)(yy)

> /{y^y^-(yy)W) J(zxY-{zz)(xx)_

^ V (yy){^z) V izz){xx)

The terms on the left are essentially positive as they repre-

sent hyperbolic cosines, those on the right are positive, being

hyperbolic sines ; we may therefore square the inequality

(xx) [yy) (zz) + 2|

{yz) (zx) (xy)\

- (asB) (yzf-{yy){zxY-{zz){xyfSO. (8)

We see that if

(yz)izx){xy)>0, (9)

we are at liberty to drop the absolute value signs in thesecond term, and the whole expression is the square of the

determinant|xyz

|which is zero or negative. We see, there-

fore, that under these circumstances,

\yz\ ^ |0a;| + |a,'2/|.

To see what region of the ultra-infinite domain is determinedby (9), let us sketch the Absolute as a conic, and draw tangentsthereunto from (y) and (z). X must lie within the quadri-lateral of these taiigents or the vertical angle at (y) or (z).

The conic and tangents determine four quasi-triangles withtwo rectilinear and one curvilinear side each. Since (yy) >our inequality (9) will hold within the quasi-triangles whosevertices are (y) and (z) and within the verticals of thesetwo angles.

Let us now assume, on the contrary, that we are in theother quasi-triangles

(yz) (zx) (xy) < 0.

Our original inequality (8) will still hold if

\xyz\^-4> (yz) (zx) (xy) < 0,( 10)


amd, conversely, this inequality certainly holds if (7) doesIt we look on {y) and (z) as fixed, and (a;) as variable, the"^"^^

\xyz\^-4>{yz){zx)(xy)=0,m so far as it lies in the two quasi-triangles we are now

+ + + + + »3 = ^+xy.

////////// JTOa+iy.

Fig. 3.

considering, will play the part of the segment of {y) and {z).*

In a region where (8) holds, a rectilinear path is the longest

from {y) to {z).

* For a complete discussion, see Study, loc. cit., pp. 103-8. Fig. 3 is takendirect.

CHAPTER VIII

THE GROUPS OF CONGRUENT TRANSFORMATIONS

The most significant idea introduced in the last chapter

was that of the Absolute, and its connexion with the concept

of distance. Every colUneation of non-euclidean space whichkeeps the Absolute in place was defined as a congruent

transformation ; we had already seen in Chapter V that every

congruent transformation was such a collineation. We maygo one step further, and say that every analytic transforma-

tion which carries the Absolute into itself alone is a congruent

transformation. Suppose that we have

(afx') = P {xx).

P must be a constant, for were it a function of {x) the

Absolute would be carried into itself, and into some other

surface P = 0, which is contrary to hypothesis. Replacing(x) by K{x) + \x. {y) we see that we shall also have

{x'y') = P{xy),

whence we may easily show that the transformation is acollineation.

It is, of course, evident, that in the complex domain, thecongruent groups of elliptic and hyperbolic space are identical,

as they are merely the quaternary orthogonal group. Inthe real domain, however, the structure of the two is quitedifferent, and our present task shall be the actual formationof those groups, pointing out besides certain interesting sub-groups. We shall incidentally treat the euclidean group as

a limiting case where tj = 0.

The group of ti'anslations of the hyperbolic line will dependon one parameter, and may be written, if fc^ = —1,

x/= iCj cosh d + Xy sin d,

i!i'= ijsinhti + ijcoshd. ^ '

We get a reflection by reversing the signs in the second

OH. VIII CONGRUENT TRANSFORMATIONS 95

equation. In the elliptic or spherical case we shall havesimilarly

x^= Xf COB d+ XiOind, .

Xi'= —Xfâind+Xicoad.

To pass to the euclidean case, replace x^, xj by kx^, kxgJ

1

and (2 by 77 > divide out Jb, and then put p = 0.

< = a!'=a;-d (3)

The ternary domain is more interesting. Let us express

the Absolute in the hyperbolic plane in the following para-

metric form

As the Absolute must be projectively transformed into itself,

we may put

<=a2,«i + a,2«2,1%I-^^"'

and this will lead to the general ternary transformation

+ 2(a„a,2+aijiaij2)a!ij,

px{= (a,i«- Og,*+ a,g2- ajjg") x^ + (oii"- aî^- a,^* + a^^) x^

+ 2(a„a,2-a2,a22)d;5s, (4)

/9a!2'= 2(anaiji + a,jaijsj)i!a + 2 (a„a2i-a,2a22)a!i

+ 2 (011022+ a21«12)*2-

If we view the matter geometrically, we see that there are

three distinct possibilities. First the two fixed points of the

Absolute conic are conjugate imaginaries. The real line con-

necting them is ultra-infinite, and has an actual pole withregard to the Absolute. This will give a rotation about this

point, and we shall have

("ii +0'-4^ = ("11-022)^+4012021 < 0.

If the fixed points of the Absolute conic be real, the trans-

formation, in the actual domain, will appear as a sliding along

a real line, if A > 0, or a sliding combined with a reflection

in a perpendicular plane through this line if A < 0. In the

third case the two fixed points of the Absolute conic fall

together, and the third fixed point of the plane falls there

too. The transformation carries a pencil of parallel lines into

itself.

96 THE GROUPS OF ch.

The elliptic case is treated similarly, by a judicious intro-

duction of imaginaries. We may write the Absolute

^^1 = ^1 ^2 1

x^ = 2^1 «2 •

Let us now take the binary substitution

fft/=(a+j3i)<i-(y + 6i)e2,

We come thus to the general group of congment trans-

formations

p<= (a^-fi'^ + y2-62) X, + 2 (y8-/3a) x^ + 2(^y -I- 6a) x^,

pxi'= 2(yh + pa)x^ + {a^-l3^-y^ + b^)x^+ 2{fib-ya)x2,

px^'=20y-ba)x^ + 2{^b + ya)x^ + {a'^ + P''-y^-b^)x2. ^'

These forms remind us at once of like forms occurring in

the theory of functions. Suppose, in fact, that we have the

euclidean sphere X^+y^ + Z^ = 1.

The geometry thereof will be exactly our spherical geometry,and we wish for the group of congruent transformations of

this sphere into itself. Let us project the sphere stereo-

graphically from the north pole upon the equatorial plane,

and, considering this as the Gauss plane, take the linear

transformation

^,^ (a + fii)z-{yÛ)^ _,^ (a-^3i)i-(y-6i)

(y— 6t)s-(-(a-/3i)' (y + 8i)2-H(a+/3i)'

These equations are seen at once to be transformable intothe others by a simple change of variables.

To pass over to the euclidean case, put

X, y,

^0 Vo

x'=C^ + A^x + B^y,

y'=C, + A^x +P- ^""f

A^B^-A^B^ = A^+B^^ = Ai^-B^ = I.

Notice that here the group

x'=Ci+x, y'=c^+y,is an invariant sub-group.

The congment groups in three dimensions are of the samegeneral form as those in two, albeit the structure is a trifle

VIII CONGRUENT TRANSFORMATIONS 97

more complicated. We wish for the six-parameter groupsleaTing invariant respectively a real, non-ruled quadric, anima^nary quadric of real equation, and an imaginary conicwith two real equations. The solution has of course, longbeen known.*The Absolute of hyperbolic space may be interpreted as

a euclidean sphere of radius one, and the problem of finding

all congruent transformations of hyperbolic space, is the sameas that of finding all collineations carrying such a sphere into

itself. Let us represent this sphere parametrically in termsof its rectihnear generators

a;, = zz+l,

Xi = zz— 1,

x^ = z + s,

x^= -i{z-z).

Let us now take the linear transformation

, az + fi ., az+ ^g

yz+ h yz + b

The six-parameter group of congruent transformations of

positive modulus will be

px^' = (aa+^^+ yy+^)Xo+ (aa-/3^ + yy-68)i!j

+ (a/3 + ayS + yS + yS) iji +i(a;3-a^ + y§-yS) Kg,

pXi' = (aa+0 + yy—^)xo+(aa~p^—yy + tl)xi

+ {a^ + afi-yl-yb)x^+i(ap-a^-yl + Yb)x3, (7)

px^' = (ay + ay + /3B+ pi) x^+ (ay + ay -_/38- /38) x^

+ {al + ab + py + Py)x2+ iiah-ab-Py + 0y)Xs,

-pXs' = i(ay-aY + pl-pb)Xa + i{ay-ay-$b+pb)xi+ i (aS— dS + /3y— /3y) ij— (aS + 58— /3y— /3y) ij

.

A = [{ab-py)(m-$y)y.

This sub-group might properly be called the group of

motions, lê total group is made up of these and the

six-parameter assemblage of transformations of negative

* The literature of this subject is large. The first -writer to express the

general orthogonal substitution in terms of independent parameters wasCayley, ' Sar quelques propri^tds des determinants gauches,' CrelU't Journal,

vol. zzzii, 1846. The treatment here given follows broadly Chapters YI andVII of Klein's 'Nicht-euklidische Oeometrie', lithographed notes, GOttingen,

1898.

COOUDOB Ct

98 THE GROUPS OF CH.

discriminant called symmetry tranaformatioîs. We reach

these latter by writing

,_ az + ^' _,_ a'z+ ^'

^ - y'z+b" ^ ~ y'z + l''

The distinction between motions and symmetry transforma-

tions stands out in clear relief when we consider the eflFect

upon the Absolute. The sub-group of motions includes the

identical transformation, and any motion may be reached bya continuous change in the six essential parameters from the

values which give the identical transformation, without ever

causing the modulus to vanish. This shows that as, underthe identical transformation, each generator of the Absolutestays in place, so, under the most general motion, the generators

of each set are permuted among one another. On the con-

traay, the most general symmetry transformation will arise

from the combination of the most general motion with areflection, and it is easy to see that a reflection wiU inter-

change the two sets of generators.

In the elliptic case we shall have the group of all real

quaternary orthogonal substitutions. An extremely elegantway of expressing these is oflfered by the calculus ofquaternions.

Let us, following the Hamiltonian notation, assume threenew symbols i, j, k:

i^ = j^ = k^ = ijk = — 1.

We assume that they obey the associative and commutativelaws of addition, the associative and distributive laws ofmultiplication. An expression of the type

is called a quaternion, whereof

I ^(^) I

is called the Tensor. It is easy to show that the tensor ofthe product of two quaternions is the product of their tensoi-s.

Let US next write

< +<! + ar/i + Xs'k = P{xg+ xî +xj + x^k) Q, <8)

where P and Q are quaternions. Multiplying out the right-hand side, and identifying the real parts and the coefficientsof i,j, k, we have x^'x(x^ x{ expressed as linear homogeneousfunctions of x^x-^x^x^. The modulos of the transformationwill be diflferent from zero, and we shall have

(a;V) = (asE).|P|^|Q|^

Till CONGRUENT TRANSFORMATIONS 99

These equations will give the six-parameter group ofmotions, the group of symmetry transformations will arise

™"^x^' + xî+x^3+Xsh =^ P' (x^-Xyi-x^j-Xsk)(^,

the distinction between motions and symmetry transformations

being as in the hyperbolic case.

Our group of motions is half-simple, being made up of twoinvariaat sub-groups GÔ^ obtained severally by assumingthat Q or P reduces to a real number. We obtain their

geometrical significance as follows :

—

The group of motions G^ can be divided into two in-

variant three-parameter sub-groups g^ g^ by resolving it into

the two groups which keep invariant all generators of the

one or the other set on the Absolute. Now were it possible

to divide Gg into invariant three-parameter sub-gi'oups in

two different ways, the highest common factor of g^ or g^'

with Cg would be an invariant sub-group, not only of G^but of gTg. This may not be, for g^ is nothing but the binaryprojective group which has no invariant sub-groups. Hencethe groups g^g/ are identical with G^ G^, and the latter keepthe one or the other set of generators all in place.

It is well worth our while to look more deeply into the

properties of these sub-groups. Let us distinguish the twosets of generators of the Absolute by calling the one left,

and the other right. This may be done analytically byadjoining a number i to our domain of rationals. Two lines

which cut the same left (right) generators of the Absoluteshall be called left (right) paratactic.* As the conjugate

imaginary to each generator of the Absolute belongs to the

same set as itself, we see that through each real point will

pass a real left and real right paratactic to each real line ; andthe same will hold for each real plane. Of course there are

possible complications in the imaginary domain, but these

need not concern us here.

Let us now look at a real congruent transformation whichkeeps all right generators invariant. Two conjugate imaginary

left generatorswill alsobe invariant,andevery fine meeting these

* The more common name for such lines is 'Clifford parallels'. Theword paratactic is taken from Study, ' Zur Nicht-euklidischen und Linien-geometrie,' Jahre^richt der deutschm llathematikervereinigung, xi, 1902. Wehave already defined parallels as lines intersecting on the Absolute, andalthough in the present case such lines cannot both be real, yet it is better to

be consistent in our terminology, especially since we shall find in ChapterXVIa, transformation carrying parallelism into parataxy. Clifford's discussion

is in his ' Preliminary Sketch of Biquatemions ', Pneeedings of the LondonMathematical Society, Tol. iy, 1873.

g2

100 THE GROUPS OF CONGRUENT TRANS, ch. viii

two will be carried into itself, every other line will be carried

into a line right poratactic to itself. Such a transformation

shall be called a left translation, since the path curves of all

points will be a congruence of left paratactic lines. In fact

this congruence will give the path curves for a whole one-

parameter family of left translations. Let the reader showthat under a translation, any two points will be transported

through congruent distances.

Eefore leaving the elliptic case, let us notice that in the

elliptic plane a reflection in a line is identical with a reflection

in a point, or a rotation through an angle ir, in a spherical

plane they are diflierent, and a reflection in a line is the sameas a rotation through an angle ir coupled with an interchangeof each point with its equivalent. In three dimensions, there

is never any identity between a rotation and a reflection, onthe other hand nothing new is brought in by interchangingeach point with its equivalent, for as each plane is herebytransformed into self, we may split up the transformationinto a reflection in a plane, a reflection in a second planeperpendicular to the first, and a rotation through an angle ir

about a line perpendicular to both planes.

To pass to the limiting euclidean case

y'=B^+B^x + B^y + B^z, (9)

where|{A^B^C^

||is the matrix of a ternary orthogonal sub-

stitution.

There will be a three-parameter invariant sub-gronp ; thatof all translations x'—A +a;

2^=A + 2/.

In like manner we may find the six-parameter assemblageof symmetry transformations.

CHAPTER IX

POINT, LINE, AND PLANE TREATEDANALYTICALLY

The object of the present chapter ie to retui-n, as promisedin Chapter VI, to the problems of elementary non-euclidean

geometry, from the higher point of view gained by extendingspace to be a perfect analytic continuum. We shall find in

the Absolute a Detus ex Machina to relieve us from many anembarrassment. We shall leave aside the euclidean case,

and, for the most part, handle all of our non-euclidean cases

together, leaving to the reader the simple task of makingthe distinction between the elliptic and the spherical cases.

Otherwise stated, our present task is to express the funda-mental metrical theorems of point, line, and plane, in termsof the invariants of the congruent group.

Let us notice, at the outset, that the piinciple of duality

plays a fundamental rdle. The distance of two points is

]c^ X logarithm of the cross ratio that they form with the

points where their line meets the Absolute, the angle of two. 1 .

planes is ^. x logarithm of the cross ratio which they form

with two planes through their intersection, tangent to the

Absolute ; the distance &om a point to a plane is -^ minus its

distance to the pole of that plane with regard to the Absolute.

Two intei'secting lines or planes which are conjugate withregard to the Absolute are mutually perpendicular. Twopoints which are conjugate with regard to the Absolute shall

be said to be mutually orthogonal. In the real domain of

hyperbolic space, if one of two such points be actual, the other

must be ideal ; the converse is not necessarily true.

Let us be^n in the non-euclidean plane, say a^ = 0. Let

us take two points A, B with coordinates («) and {y) respec-

tively, and &id the two points of theii* line which are at

102 POINT, LINE, AND PLANE ch.

congruent distances from them. These shall be called the

cerUres of gravity of the two points, and are, in fact, the twopoints which divide harmonically the given points, and the

intersections of their line with the Absolute. We purposely

exclude the spherical case, where the centres of gravity will

be equivalent points.

The necessary and sufficient condition that the point

k(x)+iJi{y) should be at congruent distances from (aj) and

The coordinates of the centres of gravity will thus be

r ^, y ^

^.^^ ^/{xx) '/(yyy

Let the reader discover what complications may arise in theideal domain.

Let us next take three non-collinear points A, B, G withthe coordinates (a;), {y), (z). A line connecting (a;) with acentre of gravity of {y) and (2) will be

V(yy)I

XxzI

+ V{zz)\Xocy

\= 0.

It is clear that such lines are concurrent by threes, in fourpoints which may be called the centres of gravity of the threegiven points. On the other hand the centres of gravity ofour pairs of points are coUinear in threes. Lastly, notice thata dual theorem might be reached by interchanging the objects,

point and line, distance and angle ; by taking, in fact, a polarreciprocation in the Absolute :

—

Theorem 1 . The centres of Theorem 1'. The bisectors ofgravity of the pairs formed the angles formed by threefrom three given points are coplanar but not concurrentcollinear by threes on four lines are concurrent by threeslines. The lines from the in four points. The pointsgiven points to the centres where these bisectors meetof gravity of their pairs are the given lines are collinearconcurrent by threes in four by threes on four lines,

points.

The centres of gravity of the points (x), (y), (z) are easilyseen to be (X y ^ \

Returning to the line BG we see that the coordinates of its

IX TREATED ANALYTICALLY 103

pole with regard to the Absolute will have the coordinates (s),

where for every value of (r)

(rs) = \ryz\.

The equation of the line connecting this point with A, i.e. theline through A perpendicular to BG, will he

{Xy){zx)-{Xz){a!y) = 0.

If we permute the letters x, y, z cyclically twice, we get twoother equations of the same type, and the sum of the three

is identically zero, so that

Theorein 2'. The points oneach of three coplanar but notconcurrent lines, orthogonalto the intersection of the other

two, are collinear.

Returning to a centre of gravity of the two points BG, wesee that a Tine through it perpendicular to the line BG will

have the equation

{xy) (xz)

(yy)J.

(yg) M ^ (gj_

/(zz)

Theorem 2. Thethrough each of three givennon-collinear points, perpen-

dicular to the line of the other

two, are concurrent.

= 0,

The first factor will vanish (in the real domain) only when

{y) and {z) are identical, the equation will then be

{^) ^=0.We see immediately from the form of this equation, that

all points of this line are at congruent distances from [y) and(z), thus confii-ming II. 33.

TheoreTn 3. If three non-

collinear points be given, the

perpendiculars to the lines of

their pairs at the centres of

gravity of these pairs are

concurrent by threes in four

points, each at congruent dis-

tances from all three of the

Theorem 3'. If three co-

planar but not concurrent

lines be given, the points

orthogonal to their intersec-

tions on the bisectors of the

corresponding angles are col-

linear by threes on four lines,

making congruent angles withall three of the given lines.given points.

Let us now suppose that besides our three original points,

104 POINT, LINE, AND PLANE GH.

we have thi-ee others lying one on each of the lines of the

first set as follows

A'={ly + mz),

B'= {pz + qx),

C'= (rx + sy).

Let us, for the moment, suppose that we are restricted to

a consistent region of the plane. Then we shall easily see

from Axiom XVI that if AA', BB', GC be concun-ent

sin-BA' . GB' . AG'

sin- sin-

< 0.

. GA' . AB' . BG'sm -rr- sin -y^ sin -r—

On the other hand, if A', B\ G' be collinear,

. BA' . GB . ACsm —T— sin —T^ sin —r--

>0.. GA' . AB' . BG'

sin —r- sin —=— sin ^—k k k

Now, more specifically, we see that

BA' 'fn''{iyy){zz)-iyzf^

whence* {yy) [t\yy) + nm{yz) + 'm\zz)'\

'

. BA' . GB' . AG'sin —j— sin —5^ sin —j—

. GA' . AR . BG'sin—i— sin —;— sin

~ \lpr)

The equation of the line AA' will be

1 1 XxyI

+mI

Xzx\= 0.

And the condition for concurrence for the three lines

{Ipr + mqs) •j

ocyz|

^ = 0,

and this will give mqa _ _Ipr

~

On the other hand, we easily see that if A', B',G' he eollinear

Ipr—Tnqs = 0.

Theorem 4. If A', B', G' be three points lying respectively


on the lines BG, CA, AB, all six points being in a consistent

region, then the expression

sin-BA' . GB'

sm- sm-BG'

sin-GA' . AB' AG'

sin -;— sin-TTk k

will be equal to —1 when, and only -when, AA', BR, GG'are concurrent, while it will be equal to 1, when, and onlywhen, A', B', C" are coUinear.

These are, of course, merely the analoga of the theoremsof Menelaus and Ceva. It is worth noticing also, that theywill afford a sufficient ground for a metrical theory of cross

ratios.

Let us next suppose that A' is a. point where a bisector

of an angle formed by the lines BA, GA, meets BG. Wefind I and m easily in this case, by noticing that A' must beat congruent distances from AB and AG, thus getting

{y V{zz)\xx)-(xzf+ z V(xx)(yy)-(ayyf),

BA' . GA' . BA . GA= sin —J— -.Bin -=~k k

sin —=— : sin ,

k k

Theorem 5. If three non-coUinear points be given, eachbisector of an angle formed bythe lines connecting two of

the points with the third will

meet the line of the two points

in such a point that the ratio

of the sines of the kth parts

of its distances from the twopoints, is equal to the corre-

sponding ratio for these twowith the third point.

Theorem 6. The locus of

a ppintwhich moves in a plane,

in such a way that the ratio

of the sines of the Arth parts

ofits distances from two points

is constant, is a curve of the

second order.

Theorem 5'. If three co-

planar but non-concurrentlines be given, each centre of

gravity of a pair of points

where two of the lines meeta third determines with the

intersection of this pair of

lines such a line, that the ratio

of the sines of the angles whichit makes with these two lines,

is equal to the corresponding

ratio for the two lines withthe third.

'Theorems'. The envelope of

a line which moves in such away in a plane, that the ratio

of the sines of its angles withtwo fixed lines is constant, is

an envelope of the second class.

106 POINT. LINE, AND PLANE ch.

It would be quite erroneous to suppose that either of these

curves would be, in general, a circle. Let the reader showthat if an angle inscribed in a semicircle be a right angle, the

eucUdean hypothesis holds.

Our next investigation shall be connected with parallel

lines. We suppose, for the moment, that we are in the

hyperbolic pkuie, and that k = i. We shall hunt for the

expression for the angle which a parallel to a given line I

passing through a point P makes with the perpendicular

to I through P. This shall be called the parallel angle of

the distance from the point to the line, and if the latter be dthe parallel angle shall be written*

n{d).

Let us give to the point P the coordinates (y), while thegiven line has the coordinates (u). Let (v) be the coordinates

of a parallel to (u) through (y). Let D be the point wherethe perpendicular to {u) through (y) meets (u). We seekcosn(d).

Since (u) and (v) intersect on the Absolute

{uu) (w)— {uvY = 0.

The equation of the line PD will be

I

xyuI

= 0.

The cosine of the angle formed by v and PD will be

V{vv) V{im){yy)-{yuf


. .J,_ sin 4-CAB ,„^

amU{AB) = sinn(£C)smn (GA) = i&n4-CABta.n4.ABC. (7)

Let the reader prove the correctness of the following con-struction for the parallels to iP-tfawH^-i^^ £ 'i_i.f.4'M\ ?*;

Drop a perpendicular from P on 2 meeting it in Q. Take Sa convenient point on the perpendicular to FQ at P, and let

the perpendicular to PS at S meet I at R. Then with P as

a centre, and a radius equal to (QB), construct an arc meetingR8 in T. PT will be the parallel required*Be it noticed that, as we should expect,

limit cos n{d) _d-0—rf—-1-

Let us now find the equations of the two parallels to theline (it) which pass through the point (y). These two cannot,naturaUj, be rationally separated one from the other, so that

we shall find the equations of both at once. Let the coordinates

of the line which connects the other intei-sections of the parallels

and the Absolute be (w). The general form for an equationof a curve of the second order through the intersections of

(u) and (vj) with the Absolute will be

I (ux) (ivx)

—

m (xx) — 0,

and this will pass through (y) if

l:m = (yy): (uy) (wy).

Since this curve is a pair of lines meeting in (y) the polar

of (y) with regard to it will be illusory, i.e. the coefficients of(x) will vanish in

(yy) {uy) {wx) + {yy) (wy) (ux)- 2 (uy) (ivy) (xy) = 0.

This last equation may be written

(wx)(wy). , >

j

(ux)iuy)

{yx){yy) ^^'^'\{yx){yy)

Now, by the harmonic theory of a quadi-angle inscribed in

a curve of the second order, w will pass through the inter-

section of (u) with the polar of y with regard to the Absolute,

so that we may write^^^ _ y^y^^ + ^y^^

{ux) {uy)

(uy) = 0,

Substitutingt2,(,,)^^(,,)] W (yy)

= 0.

* The formulae given may be used as the basis for the whole trigonometricstmctnre. Cf. Uanning, Nan-eudidean Geometry, Boston, 1901. Manning'sreasoning is open to very grave question on the score of rigour.


The coefficients of x^x^x.^ will vanish if

^= -{yy)> M = 2(u7/).

Under these circumstances

(vxc) = -(2/2/) (ux) + 2 (uy) (xy),

{ivy) = {yy){'wy).

Which leads to the required equation

(uyY (XX) + (uxf {yy)-2{ux) {uy) {xy) = 0. (8)

To get the euclidean formula, replace oj^ by k^x^ and divide

by k. We get the square of the usual expression

[{uy)x,-{ux)y,Y = 0. (9)

The principles which we have followed in studying the

metrical invariants of the plane may be extended with ease

to thi-ee dimensions. We have merely to adjoin the fourth

homogeneous point or line coordinate.

Let us have four points, not in one plane, with the coor-

dinates {x), {y), {z), {t) respectively. We easily see that theeight points

(-^ + -4^ + -J= + -i=), (10)^^/{xx}- V(yy)~ </{zz)~ "/{tty

will be points of concurrence, four by four, of lines from eachof the given points to the centres of gravity of the other three.

These eight may, in fact, be called the centres of gravity of thefour points. The centres of gravity will form with the givenpoints a deamic configuration.* The meaning of this phraseis as follows. Let us indicate the centres of gravity by thesigns prefixed to their radicals, giving always to the first

radical a positive sign. We may then divide our twelvepoints into three lots as follows :

—

(^) {y) (2) m(+ + + +)(+ + --)(+- + -)(+--+) (11)

(+ + + -)(+ + -+)(+- + +)(+ )

We see that a line connecting a point of one lot, with anypoint of a second, will pass through a point of the third. Thetwelve points will thus lie by threes on sixteen lines, four

* The desmic configuration was first studied by Stephanos, ' Sur la con-figuration desmique de trois tetraddres,' Bulletin des Sciences malhematimes,B^rie 2, toI. iii, 1878.


pasBing through each. In like manner we shall find that if

we take the twelve planes obtained by omitting in turn onepoint of each lot, two planes of different lots are always coaxalwith one of the third. Let the reader who is unfamiliar withthe desmic configuration, study the particular case (in euclidean

space) of the vertices of a cube, its centre, and the ideal pointsof concurrence of its parallel edges.

Theorem 7. If four non-coplanar points be given, the

lines from each to the four

centres of gravity of the other

three will pass by fours

through eight points whichform, with the original ones,

a desmic configuration.

Theorem 7'. If four non-concurrent planes be given,

the lines where each meetsthe planes which severally are

coaxal with each of the three

remaining planes and a planebisecting a dihedral angle ofthe two still left, lie by fours

in eight planes which, withthe original ones, form adesmic configuration.

Let the reader show that the centres of gravity of the six

pairs formed from the given points will determine a second

desmic configui'ation, and dually for the planes bisecting the

dihedral angles.

Let us seek for a point which is at congruent distances

from our four given points. It is easy to see that there cannotbe more than eight such points. Their coordinates are foundto be (s) where, for all values of r,

(ra) = v^(axc)|ryzt

\+ V{yy)

|mix

\± -/{zz) \

rtxy\

±-/(tt)|ra^3|. (12)

Theorem 8. If four non-coplanar points be given, the

eight points which are sever-

ally at congruent distances

from them form, with the

original four, a desmic con-

figuration.

Theorem 8'. If four non-concuiTent planes be given,

the eight planes which sever-

ally meet them in congruentdihedral angles, form, with theoriginal four, a desmic con-figuration.

As there are eight points at congruent distances from the

four given points, so there will be eight planes at congruent

distances from them, we have but to take the polars of the

eight points with regard to the Absolute. In like manner,if we consider not the points (x), {y), {z), (t) but their four

110 POINT, LINE, AND PLANE OH.

planes, there will be eight points at congruent distances from

them. The coordinates of these latter eight will be

±y^0 ^1 ^2 -3

to tl k tz

Theorem 9. If four non-

coplanar points be given, the

eight points which, severally,

are at congruent distances

from the planes of the first

four, form, with the first four

points, a desmic configura-

tion.

2^0 2l ^2 ^3

^0 '^l ^2 '3

1 ' 3

to ti t, t.


Each will meet the Absolute of polar of the other if

^PijP'ij = 0- (16)

Notice that (p |

p') is an invariant under the general group of

collineations, while ^PijP'ij is invariant under the congruentgroup only.

We shall mean by the distance of two lines the distance

of their intersections with a third line perpendicular to themboth. It is easy to see that if two lines be not paratactic,

there will be two lines meeting both at right angles, and these

are indistinguishable in the rational domain, that is, in the

general case. If, thus, d be taken to indicate the distance

of two lines, sin^ t will be a root of an in-educible quadraticfC

equation, whose coefficients are rational invariants under the

congruent gi'oup. Let us seek for this equation.

iSt one of our lines be p given by the points (as), (y), while

the other is (p') given by (a;') and (y'). For the sake of

simplifying our calculations we shall make the obviously

legitimate assumptions

{xy) = (mf) = (x'y) = (xY) = 0.

The distances which we wish to find are

k

We have

d^ _ ^(xx)(x'x')-(x!>/)\,^d,_ V(yy)(y'y)-(yy'y

v^) V(2/y)sm-r = sin-;^ =

V(xx) >/(a^x') ~ k

{xx){yy)-{xyf = '2,pij\

and this will vanish only when {p) is tangent to the Absolute,

a possibility which we now explicitly exclude both for {p)^°'^(^')-

{xx){yy) = 'Lpi^\ (a=V)(^y) = SpV'

p'Y = I

xy x'y'I

^

(axe) (xx')

(yy) (yy')

(asc') (a/a^)

{yy') (y'y')

= [(xx){x^x')-(xxfy] [{yy){y'y')-(yy'n

(P

. ,d, . „ tZj

sin* -r sin^ t= =k

Bin' -^ Bin* -r = 'd,

k

[{xx) (a;V)- {xx'f] \(yy) (y'y')- (yy')^

{xx){x'x') {yy){y'y')

SPi/W"

(17)

(18)


sin''^ sin''-* - 1-cos''^ -cos»-^ + (^^ZMl,

^PvPv \{yx'){yy')= (a;«') {yy'},

k k Spij Spip

(p\prsini^-J +sin^^= 1 ^—

^

d^ =

T^-k-^ ^Pij^^Pi/^

2pi/ 2 p,/^ sin* ^ + [{2pij pi/f- (p I p'f- S^,/ 2 pi^'^] sin*^

+(p|/)'»= 0. (19)

^Pi;-^Pi/'<ô^'^+i(p\pr-0PijPi/f-^Pi/PiP'\'^°^'t

+ i^PijPi/)'^0. (20)

The squaxe roots of the products of the roots of these twoequations are well-known metrical invariants, and have beenstudied under the names of moment and commoment of thetwo lines.* We shall return to the moment presently, attach-

ing a particular value to the signs of the radicaJs in the

denominator. If two lines intersect the moment must be zero,

and if each intersect the absolute polar of the other, thecommoment must vanish, thus bringing us back to equa-tions (15), (16).

To reach the limiting euclidean case we replace, as usual,

Xg by kxg, divide out k\ and put -p = 0. Then, since

limT . d J

, ffsm-r = a.

We have

iPlpJ{Poi +P^ +P(a) (Poi" +Poi''+Po3")-(PoiPai' +PoiPoi'+PmPiaT

the usual formula. \^^'

With regard to the signs of the roots in (19) we see that in

the hyperbolic case, where the two lines are actual, one of

* See D'Ovidio, 'Studio sulla geometria proiettira,' Annali di Matematiea,vi, 187S, and ' Le fanzioni metriche fondamentali negli spazii di quantesi-Togliono dimensioni ', tfemorte dei Lituei, i, 1877.

sin-


the points chosen to determine each line will be actual and theother ideal-, so that

(p\pr<o,

Bin*^'sin*$<0.k k

The square of the moment of the two lines is negative, so that

one distance will be real and the other pure imaginary. Inthe elliptic case the two distances will be real.

We shall mean by the angle of two non-intersecting lines

the angles of the plane, one through each, which contain the

same common perpendicular. This will be k times the corre-

sponding distance of the absolute polars of the lines. Wethus get for the angles 6 of the two lines (/>), (p')

^Pi/^Pi/' ^^*0+[{lpijPi/f-{p \p'f-^Pif ^Pi/"^-] sin'^fl

+ (p\p'f = 0.

To get the euclidean formula we make the usual substitu-

tions and divisions, and put z = 0, thus getting the well-

known formula

.. (a;i'+V+ ara") {x{^ + x^^ + x^'^)-{x^x^ +xX + <^z^zf /go"

{x{' +V + x^') {x{-^ -H <=> + x{')' ^^''

The coordinates of the line q cutting p and p' at right angles

will be given by

{p\q) = {p'I q) = ^Pij qij = 2 Pif q^j = {q\q) = 0.

We have defined as a parallel, two lines whose intersection

is on the Absolute ; let us now give the name pseudoparalld

to two coplanar lines whose plane touches the Absolute. Thenecessary and sufficient condition that two lines should be

either parallel or pseudoparallel is that they should intersect,

and that there should be but a single line of their pencil

tangent to the Absolute. These conditions will be expressed

by the equations

ip I

p') = [S^,/ ^Pi/'-{^PijPi/n = 0. (23)

Let the reader notice that when we pass to the limit in the

usual way for the euclidean case, our equations (23) become

{p\p') = Bme = 0. (24)

Let us now look at paratactic lines, i.e. lines which meetthe same two generators of one set of the Absolute. Of course

114 POINT, LINE, AND PLANE en.

it is in the elliptic case only that two such lines can be real.

It is immediately evident that two paratactic lines have aninfinite number of common perpendiculars whereon they

always determine congruent distances, we have, in fact,

merely to look at the one-parameter group of translations

of space which carry these two lines into themselves. Con-

versely, suppose that the distances of two lines be congruent.

Besides our previous equations connecting {x)(y){oif){y'),yTQ

^^^^{xx'f ^ (yyy

{XX) (x'x') iyy) (y'y')'

The lines p, p' meet the Absolute respectively in the points

(x-Z^) ± iy -/(xx)) ix' •/(yV) ± i'tf Vix'x')).

It is clear, however, that every point of the line

{xv^) + iy VJxx)) {x'VW7) + iy' -/(x'a^)),

and of the liae

(xy/{yyj-iyV{xxj) (x'V^y^j—iyWlxx)),

belongs to the Absolute; the lines are paratactic. Lastly,the absolute polars of paratactic lines are, themselves, pai-a-

tactic. Hence

Theorem, 10. The necessary and sufficient condition thattwo lines should be paratactic is that their distances or anglesshould be congruent.

This condition may be expressed analytically by equatingto zero the discriminant of either of our equations (19), (20).

{i(p\p')+{^PijPij')Y-'^Pif^PiP}{[{pW)-{'iVijPi/)y

-^Pi/^Pij"}=0. (25)

This puts in evidence that intersecting lines cannot beparatactic unless they be parallel, or pseudoparallel.

In conclusion, let us return for an instant to the moment oftwo real lines, , , , , ,.

smV sm -T^ = — ^ '-^ '.

* A îpi/ ^2pi/'We shall assume that the radicals in the denominator

are taken positively, so that the sign of the moment isthat of (p I

p'). We now proceed to replace our concept ofa line by the sharper concept of a ray as follows. Let us,


in the hyperbolic case assume always »„ > 0, and in theelliptic case x^ > 0. The coordinates

Pi Pii =ViVj

shall he called the coordinates of the ray from {y) to {z), andthis shall he considered equivalent to any other ray whosecoordinates differ therefrom by a positive factor. Inter-

changing {y) and (a) will give a second ray, said to be opposite

to this. The relative moment of two rays is thus determined,both in magnitude and sign. We shall later

applications of this concept.

see various

h2

CHAPTER X

'

THE HIGHER LINE-GEOMETRY

In Chapter IX we took some first steps in non-euclidean

line-geometry. The object of the present chapter is to

continue the subject in the special direction where the

fundamental element is not, in general, a line, but a pair

of lines invariantly connected.*

Let us stai-t in the real domain of hyperbolic space and

consider a linear complex whose equation is

(d|p) = 0.

The dots indicate that the coordinates of a point are

x^, Xi, x^, X3, and choosing such a unit of measure that

k^ = — 1, we have for the Absolute

-x„^ + Xi^ + x^'' + x^'' = 0.

The polar of the given complex will have the coordinates

«oi = ^^jfc> "jfc = ~ ''*oi' *• 3> * = 1, 2, 3,

and the congruence, whose equations are

(a1 i>)

= 2 aoi Pot- 2 ajfc Pjk = 0.

will be composed of all lines of our complex and its absolute

polar, or common to all complexes of the pencil

{l&oi- mdja) . . . (Zttas + î)-These complexes shall be said to form a coaxal pencil, and

the two mutually absolute polar lines, which are the dkeotrices

of the congruence, shall be called aoces of the pencil. We get

their pliickerian coordinates by giving to i : m such values

that the complex shall be special. Let us now write

«01 + *"23 = P^\>

* Practically the whole of this chapter is sketched, without proofs, byStudy in his article, 'Zur nicht-euklidischen etc.,' loc.cit. The elliptic case

is developed at length in the author's dissertation, 'The dual projective

geometry of elliptic and spherical space,' Oreifswald, 1904. For the hyper-bolic case, see the dissertation of Beck, ' Die Strahlenketten im hyperbolischenRaume,' Hannover, 1905.

CH. X THE HIGHER LINE-GEOMETRY 117

A complex coaxal with the given line will be obtained bymultiplying the numbers (Z) by {l + mi).A pair of real lines which are mutually absolute polar,

neither of which is tangent to the Absolute, shall be called

a proper cross. They will determine a pencil of coaxalcomplexes. If either of the lines have the pluckerian coor-

dinates (a), then the three numbers (X) given by equations (1)

may be taken to represent the cross. These coordinates (X)are homogeneous in the complex (i.e. imaginary) domain, for

the result of multiplying them through by {l + mi) is to

replace the complex (d) by a coaxal complex, and therefore

to leave the axes of the pencil unaltered.

Conversely, suppose that we have a triad of coordinates (X)which are homogeneous in the imaginary domain. The coor-

dinates of the lines of the corresponding cross will be foundfrom (1) by assigning to p such a value that the coordinates

(a) shall satisfy the fundamental pluckerian identity. Forthis it is necessary and sufficient that the imaginajy part ofp^ {XX) should vanish, i.e.

cdo, = C-^^ + -^^=VV(ZZ) Vixxl^2j

./ Xf Xi \

To get the other line of the cross, i.e. the Absolute polar

of the line (d), we merely have to reverse the sign of one

of our radicals.

There is one, and only one case, where our equations (2)

become illusory, namely where

{XX) = 0.

This wiU arise when(d|d) = 2doi''-Sd..fc2 = 0,

i.e. when the directrices of the congruence are tangent to the

Absolute. All complexes of the pencil will here be special,

and will be determined severally by lines intersecting the

various tangents to the Absolute at this point. Any mutually

polar lines of the pencil of tangents, will, conversely, serve to

determine the coaxal system. We may then represent such

a pencil of tangent lines by a set of homogeneous values {X)

where {XX) =0, and, conversely, every such set of homo-

geneous values will determine a pendl of tangents to the

Absolute. We shall therefore define such a pencil of tangents

as an improper, cross.

118 THE HIGHER LINE-GEOMETRY ch.

Tîeorem 1. There exists a perfect one to one correspondence

between the assemblage of all crosses in hyperbolic space, and

the assemblage of all points of the complex plane of elliptic

space. Improper crosses will correspond to points of the

elliptic Absolute.

We shall say that two crosses intersect if their lines inter-

sect. The N. S. condition for this in the case of two proper

crosses will be

{XT) ^ ±(XY)

V{XT) ^(YT) ^(ir) v'^fF)'

Geometrically a line may intersect either member of a cross.

This ambiguity disappears in the case of perpendicular inter-

section.

Theorem 2. Two intersecting crosses will correspond to

points, the cosine of whose distance is real, or pure imaginary

;

crosses intersecting orthogonally will correspond to orthogonal

points of the elliptic plane.

The assemblage of crosses which intersect a given cross

orthogonally will be given by means of a linear equation.

A linear equation will be transformed linearly into anotherlinear equation, if the variables and coefficients be treated

contragrediently. Geometrically we shall imagine that ourassemblage of crosses, cross space let us say, is doubly over-

laid, the crosses of one layer being represented by points andthose of the other by lines in the complex plane, we have then

Theorem 3. The necessary and sufficient condition that twocrosses of different layers should intersect orthogonally is that

the corresponding line and point of the complex plane shouldbe in united position.

If a cross be improper, the assemblage of all crosses cuttingit orthogonally will be made up of all lines through the pointof contiict, and all lines in the plane of contact. This assem-blage, reducible in point space, is irreducible in cross space.

The coUineation group of cross space, is the general groupdepending on eight complex, or sixteen real parameters

pXi'=±aijXi, lUijlÔ. (3)

i

When will this indicate a transformation of point space?It is certainly necessary that improper crosses should go intoimproper crosses, hence the substitution must be of the ortho-

X THE HIGHER LINE-GEOMETRY 119

gonal type. Moreover, the Absolute of hyperbolic space willbe transformed into itself, so that our transformation of pointspace must be a congruent one. Conversely, it is immediatelyevident that a congruent transformation will transform cross

space linearly into itself. Also, an orthogonal substitution in

cross coordinates wUl carry an improper cross into an im-proper cross, and will carry intersecting crosses into other

intersecting crosses. The corresponding transformation in

point space is not completely determined, for a polar recipro-

cation in the Absolute of point space appears as the identical

transfoi-mation of cross space. A transformation whichcan-ies intersecting crosses into intersecting crosses may thus

be interpreted either as a collineation, or a correlation of

point space.

Theorem 4. Every collineation or correlation of hyperbolic

space which leaves the Absolute invariant will be equivalent

to an orthogonal substitution in cross space, and every suchorthogonal substitution may be interpreted either as a con-

gruent transformation of hyperbolic space, or a congruent

transformation coupled with a polar reciprocation in the

Absolutei

Let us now inquire as to what are the simplest figures of

cross space. The simplest one dimensional figure is the chain

composed of all crosses whose coordinates are linearly depen-

dent, by means of real coefficients, on those of two ^vencrosses, pXi = aYt + bZi, i = 1, 2, 3. (4)

Interpreting these equations in the complex plane we see

that we have co^ points of a line so related that the cross ratio

of any four is real. If this line be represented in the Gauss

plane, the chain will be represented by a circle. If the line

be imaginary, the real lines, one through each point of the

chain, will generate a linear pencil or a regulus.*

The crosses of the chain will cut oithogonally another cross

(of the other layer) called the axis of the chain. The axis

being proper, the chain will contain two improper crosses,

namely, the pencils of tangents to the Absolute where it

meets the actual line of the chain.

There is a theorem of very great generality connected with

chains, which we shall now give. Suppose that we have a

* The concept ' chain of imaginary points ' is due to Yon Staudt. See his

' Beitrftge' loc. cit, pp. 187-42. For an extension, see Segre, ' Su un nuovo

campo di ricerche geometriche,' Alii della S. Accademia delle Scieme di Torino,

vol. XXV, 1890.

120 THE HIGHER LINE-GEOMETRY CH.

congruence of lines of such a nature that the correeponding

cross coordinates (U) are analytic functions of two real

parameters u, v. The cross of common perpendiculai"8 to the

cross {U) and the adjacent cross (U+dU) will be given by

(5)

^j f^fc


If (ps— gr) = or {pr+ q8) = 0,

we have Wo real and two imaginary linear pencils ; the con-ditions for this in cross coordinates will be invariant underthe orthogonal, but not under the general group. The generalform of our surface is a ruled quaiiic, having a strong simi-

larity to the euclidean cylindroid.

The simplest two dimensional system of crosses is the chaincongruence. This is made up of all crosses which have coor-

dinates linearly dependent with real coefficients on those of

three given crosses which do not cut a fourth orthogonally

\XTZ\^0, i = l,2, 3. (9)

Theorem 6. The crosses which correspond to the assemblageof all points of the real domain of a plane will generate

a chain congruence.

Theorem 7. The common perpendiculars to pairs of crosses

of a chain congruence will generate a second chain congruencein the other layer. Each congruence is the locus of the axes

of the 00^ chains of the other ; the two are said to be reciprocal

to one another.

The reciprocal to the chain congruence (9) will have equa-

tions ]TiY.\IZ. Zj, r, T.

Z^Z^\ -\T^T^£r,. = 2J * * -^g

"îÛ(10)

Let the reader show that the chain congruence may be

reduced to the canonical form

Zi = a(p+ g*), Z2 = 6(r+8i), Z3 = c(f-Hri),

where a, h, c ai-e real homogeneous variables.

There are various sub-cases under the congruent group. If

{•l^-qr) = 0,

the congruence will be transformed into itself by a one-

parameter group of rotations.

Again, let (^_gr) = 0, (pr-qt) = 0.

Here we see that (XX')

y{ZX) -/(FZOis real for any two crosses of the congruence, i.e. the con-

gruence consists in all crosses through the point (1, 0, 0, 0).

Leaving aside the special cases the following theorems maybe proved for the general case.

Theorem 8. The chain congruence, conmdered as an assem-

122 THE HIGHER LINE-GEOMETRY CH.

blage of lines in point space, is of the third order and class.

It is generated by common perpendiculars to the pairs of lines

of a regulus. Those lines of the congruence which meet a line

of the reciprocal congruence, orthogonally generate a quartic

surface, those which meet such a line obliquely generate a

regulus whose conjugate belongs to the reciprocal congruence.

The two congruences have the same focal surface of order and

class eight.

Another simple two-parameter system of crosses is the

following

pTi + qZi + aTiÔ, \YZT\ = 0, {ahcpqr) real.

All these crosses cut orthogonally the cross

U,=

Conversely, let us show that every cross orthogonally inter-

secting (U) may be expressed in this form. As such a formas this is invariant for all linear transformations, we maysuppose Y^ = Z^ = T^ = 0.

We have then the equations

aFi + hZ^ + cT^ = (r+ iV) X^,

aT^ + bZ^ + cT^ = (r +i/) X^,

which amount to four linear homogeneous equations in five

unknowns a, h, c, r, / and these may always be solved. Therewill be found to be one singular case where the same cross

has co' determinations.

The assemblage of crosses cutting a cross orthogonally is

but a special case of what we have already defined as asynectic congruence. If

azaxX = X{uv) I

iu iv= 0,

there will be but one common perpendicular to a cross and its

adjacent crosses. This corresponds to the fact that there willexist an equation

ff^x^ X^ X^) = 0,

so that our congruence is represented by a curve, the tangentat any point representing the common perpendicular justmentioned (in the other layer), and, conversely, every curvewill be represented by a synectic congruence. The points andtangents will be represented by two synectic congruences so


related that each cross of one is a cross of striction of a crossof the other, and all its adjacent crosses. We may reacha still clearer idea of these congruences by anticipating someof the results of differential geometry to be proved in later

chapters. For, if we look upon the congruence of lines

generated by our crosses, we see that the two focal points

on each are orthogonal and the two focal planes mutuallyperpendicular. From this we shall conclude that our line-

congi-uence is one of normals, and the characteristics of the

developable surfaces of the congruence will be geodesies of

the focal sui-&ce, to which the lines of the other congruenceare binormals. We shall, moreover, show in a later chapterthat if rj and r^ be the radii of curvature of normal sections of

a surface in planes of curvature, then the Gaussian expression

for the curvature of the surface at that point will be

1 1}^

Atan-r ^tan-rk k

In the present instance as the two focal points are orthogonal

r„ Tc r, 1 1 1 _

k 2 k , . r,J , r, k^

k k

Our congruence is made up of normals to surfaces of Gaussian

curvature zero, i.e. to surfaces whose distance element maybe written d8^ = du^ + d^.

Theorem, 9.* A synectic congruence will represent the points

of a curve of the complex plane. It will be made up of crosses

whose lines are normals to a series of surfaces of Gaussian

curvature zero. The characteristics of the developable surfaces

are geodesies of the focal surfaces. Their orthogonal trajec-

tories are a second set of geodesies whose tangents will

generate a like congruence.

In conclusion, let us emphasize the distinction between

these congruences and the non-synectic ones, where the

common perpendiculars to a cross and its adjacent ones

generate a chain.

Did we wish to represent the imaginary as well as the

real members of a synectic or non-synectic congruence, weôuld be obliged to introduce into our representing plane,

points with hypercomplex coordinates. We shall not enter

into this extension, for, after all, the real point of interest of

* Cf. study, ' Zur nicht-euklidischen etc.,' cit., p. 328.

124 THE fflGHER LINE-GEOMETRY ch.

the subject lies merely in this, namely, to give a real inter-

pretation for the geometry of the complex plane.

As we identify the geometry of the cross in hyperbolic

space with that of a point of the complex plane, so we mayrelate a cross of elliptic (or spherical) space to a pair of real

points of two plane. The modus operandi is as follows :

—

We start, as before, with a pencil of coaxal linear complexes

defined by„.„ xr _ „_„t-

«03 + «12 = Pl^3' «03— "l2 = <^T^3-

If we replace our complex by another coaxal therewith, weshall merely multiply dX) (,Z) by two different constants.

Conversely, when we wish to move back from the indepen-

dently homogeneous sets of coordinates (jX) (^X) to the

degenerate complexes of the pencil, i.e. to the lines of the

cross defined thereby, we have to take for p and a such values

that the fundamental pliickerian identity is satisfied,

ra,i = jX, y(X^) + rî>/W^ ,^2.

rCljU = lî Ar^r^)-rîAlXlX).The two separately homogeneous coordinate triads (jX) (^)

may be taken to represent this proper cross, and, conversely,

as all quantities involved so far are supposed to be real, everyreal pair of triads will correspond to a single cross.

Theorem 10. The assemblage of all real crosses of elliptic

or spherical space may be put into one to one correspondencewith the assemblage of all pairs of points one in each of tworeal planes.

Our doubly homogeneous coordinates have a second inter-

pretation which is of the highest interest. Let us write thecoordinates of a point of the Absolute in terms of two inde-pendent parameters, i.e. of the parameters determining theone and the other set of linear generators

= (îMi-^2/*2) :(îMi + V2) : (îM2 + ^2Mi): (îM2-^2Mi)-

The pliickerian coordinates of a generator of the left or rightsystem will thus be

i'oi =P2& = ^XjAj, goi = -ffss = ^Htî,

Poi = Psi = *(V + ^2*). 902 = - ?31 = *W + M2*).

P03 = ^12 = (}^l - V). 308 = -2l2 = -W -lî)-


The parameter (X) of a left generator which meets a givenline (o) will satisfy

2AiA2(a„i + a^)+i(V+X/)(a^ + a3,) + (V-V)(«o3+ ai2) = 0-

Similarly, for a right generator we have

2Mif*2Ki-«23) +i(Mi*+/*,') (ao2-03i)-fMi''-/*2^) (ao3-«i2)= 0.

We thus get as a necessary and sufficient condition thattwo lines should be right (left) paratactic, that the differences

(sums) of complementary pairs of pliickerian coordinates inthe one shall be proportional to the corresponding differences

(sums) in the other. If the lines be (p) and {p'), the first ofthese conditions will be

lip \p')+'^PijPi/V-'^Vi^^PiP = 0.

while the second is

{{P \p')-^PijPii]'-^Pi/ ^Pij' = 0.

If these equations be multiplied together, we get (25) ofChapter IX.

If a line pass through the point (1, 0, 0, 0) its last three

pluckerian coordinates will vanish, while the fii'st three are

proportional to those of its intersections with Xg = 0. It thusappears that in (11) the coordinates dX) and (^X) are nothingmore nor less than the coordinates of the points, where the

plane a;o = is met respectively by the left and the right

paratactic through the point (1, 0, 0, 0) to the two lines of the

cross, for a line paratactic to the one is also paratactic to

the other. It will, however, be more convenient to consider

(iX) and {rX) as standing for points in two different planes,

called, respectively, the left and right representing planes.

We shall speak of two crosses as being paratactic, when their

lines are so, and the necessary and sufficient condition there-

fore, invariant under the group of cross space, is that theyshould be represented by identical points in the one or the

other plane.*

As in the hyperbolic case, so here, we shall look upon cross

space as doubly overlaid, and assign a cross to the upper layer

if it be determined by two points in the representing planes,

while it shall be assigned to the lower layer if it be determined

by two lines. Under these circumstances we may say :

—

Theorem, 11. In order that two crosses of different layers

should intersect orthogonally, it is necessary and sufficient

* The whole question of left and right is considered most carefully in

Study's 'Beitrage', cit., pp. 126, 156.

126 THE fflGHER LINE-GEOMETRY ch.

that they should be represented by line elements in the two

planes.

We may go still further in this same direction. We shall

mean by the right and left Clifford angles of two crosses, the

angles of right and left paratactics to them through any chosen

point Let the reader show that the magnitude of these angles

is independent of the choice of the last-named point. If, thus,

we choose the point (1, 0, 0, 0), the cosines of the CliflFord

angles will be

yGzir)^GFjF)' V(;;j;z) AF,F)Now, from equations (19) and (20) of Chapter IX, we see that

. d . d' . . . ^ {p\p')sm^sm^=sm0sm^=-^_-^_.d d' a Of ^PiiVii

cos -r COS -r = cos 9 cosfl'=k k

hence, we easily find

V^Pif v^S^"

(d d\ _ iiXiY)

\k k)~cos

or else

yd ^ d\ iiXiY)

AiXiX) V(jr,F)

The ambiguity can be removed by establishing certain con-

ventions with regard to the signs of the radicals, into whichwe shall not enter.* We may, however, state the following

theorem :

—

Theorem, 12. The Cliflford angles of two lines have the samemeasures as the sums and differences of the lc\h parts of their

distances, or the sums and differences of their angles. Thenecessary and sufficient condition that two lines should inter-

sect is that their Clifford angles should be equal or supple-

mentary.

* For an elaborate discussion, see Study, ' Beitr&ge,' cit., especially p. 130.

5c THE HIGHER LINE-GEOMETRY 127

When we adjoin the imaginary domain to the real one,Berions complications -will arise which can only be removedby careful definition. Without going into a complete dis-

cussion, we merely give the facts.*

If (jZjZ) = 0, (,X,Z):^0, we shall say that we havea left improper cross, and denote thereby a left generatorof the Absolute, conjoined to a non-parabolic involution

among the right generators. There will be oo* such impropercrosses, and oo' right improper crosses, whose definition is

obvious. Left and right improper crosses together will con-stitute what shall be called improper crosses of the first sort.

Improper crosses of the second sort shall be defined, as in

hyperbolic space, as pencUs of tangents to the Absolute,

corresponding to sets of values for which (jXjX)= (,Z,X)= 0.

The definitions of parataxy and orthogonal intersection maybe extended to all cases, their analytic expression being as

in the real domain.The general group of linear transformations of cross space

will depend upon sixteen essential parameters. It will bemade up of the sixteen-parameter sub-group Oig of all trans-

formations of the type

PjZ/=2«yjî. <^rî=1bij,Xj, \aij\x\bij\^0, (14)

and the sixteen-parameter assemblage H^g of all transforma-

tions of the type

PjX/=2«y-r^j. «^r^'= 2^y-Iî' I «y-M ^j I^ 0. (15)

; i

Notice that under G^^ left and right parataxy of crosses of

the same layer are invariant, while under Sj^ the two sorts

of parataxy are interchanged.

The group G^g will contain, as a sub-group, the group of all

motions, while H^g includes the assemblage of all symmetrytransformations. Let the reader show that there can be nocollineations of point space under G'lg, except congruent trans-

formations, and that the necessary and sufficient condition

that (14) should represent a motion of point space is that

the transformations of the two representing planes should

be of the orthogonal type.

The group G^ is half-simple, being composed entirely of

two invariant sub-groups jCg, ,Gg, of which the former is

made up of the general linear transformation for (jZ) with

* Cf. the author's 'Dual projective Geometiy', loc cit., § 3.

128 THE HIGHER LINE-GEOMETRY ch.

the identical transformation for (,X), while in the latter, the

rdles of (jX) and (^) are interchanged. The highest commonfactors of the group of motions with jGg and ^Gg respectively,

will be the groups of left and right ti-anslations (cf.

Chapter IX).

The simplest assemblages of crosses in elliptic space bear

a close analogy to those of hyperbolic space, although pos-

sessing more variety in the real domain. Let

iXi = aiYi + hZi, î = a,Yi + 6,Z<,

The assemblage of crosses so defined shall be called a chain.

The properties of these chains are entirely analogous to those

in the hyperbolic case. For instance, take a congruence of

crosses whose coordinates are analytic functions of two essential

parameters (u), (y). Let us further assume that (jF) (,F)

being crosses of the system

rl F- F *o-

The meaning of this restriction is that neither (jF) nor (^F)

can be expressed as functions of a single parameter, so that

the crosses of the congruence cannot be assembled into the

generators of oo^ surfaces, those of each surface being para-

tactic. Let the reader then show that for every suchcongruence, the common perpendiculars to a line in the

general position, and its immediate neighboui-s, will generate

a chain.

The chains of elliptio cross space wUl have the same sub-classifications under the congruent group, as in the hyperbolicplane. Let the reader show that the general chain may berepresented by means of a homographic relation between the

points of two linear ranges in the representing planes, andthat the special chain, composed of two pencils, arises, whenthe relation is a congruent one.

Suppose, next, that we have

PlXi = aiYi + biZf, <r,X, = a,Yi + bî,

This is a new one-parameter family of crosses called a strip,

or, more exactly, a left strip. The common perpendicularsto pairs of crosses of tiie left strip will generate a right strip

(whereof the definition is obvious), and each strip shall be said

to be reciprocal to the other. A left strip of the upper layer


vrill be represented by a point of the left plane, and a linear

range of the right plane. The reciprocal strip in the lowerlayer will be represented by the pencil through the pointin the left plane, and the line of the range in the right.

In point space, the lines of a stnp are generators of aquadric, whose other generators belong to the reciprocal strip.

Owing to the parataxy of the generators of such a quadric,

it wiQ intersect the Absolute in two generators of each set.

We shall call our quadric a Clifford m/rfcbce, when we wishto refer to it as a figure of point space. We shall show in

Chapter XV, that these surfaces have Gaussian curvature zero,

since they are generated by paratactic lines, and are minimalsurfaces, since their asymptotic lines form an oi'thogonal

system.*The simplest two dimensional system of crosses will be, as

before, the chain congruence

iXi = aiYi + h^Zi + ciTi, ,Xi = aiXi + b^Yi + c^Z^

\lYiZ{£\ X \,Y,Z,T\i^Q.

We may solve the first three equations for a, 6, c, and sub-

stitute in the last

This, again, may easily be reduced to the canonical foi-m

,X, = a,.jXi. (16)

The reciprocal congruence will be given by

There are various sub-classes under the congruent group.

If the squares of no two of our quantities a^ in (16) be equal,

we have the general congruence, if we have one such equality,

the congruence will be transformed into itself by a one-

parameter group of rotations. If all thi'ee squares be equal,

we have a bundle of crosses through a point. The general

congruence will have all of the properties mentioned in (8).

A different sort of congruence will arise in the case where

\,Y,Z{r\ = % UF.Z.TjÔ. (17)

This congruence will contain oo' strips, whose reciprocals

generate the reciprocal congruence. The common perpen-

diculars to all non-paratactic crosses of the congruence will

generate a bundle, those to paratactic crosses, the reciprocal

* Cf. Klein, ' Zur nicht-euklidischen Oeometrie,' HaihemaMsche AtxwHen,

vol. xxxvii, 1890.

COOLlDQi: I

130 THE HIGHER LINE-GEOMETRY CH. x

congruence. Such a congruence wiU be generated by the

common perpendiculars to the paratactic lines of two pencils

which have different centres and planes, but a common line

and paratactic axes. In point space the line congruence wiUbe of order and class two. The canonical form will be *

iX, = 0.

If, in addition to (17), we require the first minors of|

lYiZiT|

all to vanish, we shall have a bundle of paratactic crosses.

If, on the other hand, we have

without the vanishing of the first minora of either determinant,

we have oo^ crosses cutting a given cross orthogonally. Theequations of the congruence may be reduced to the canonical

fo™PiX^ = a, .T,X, = b,

Pl^2 = ^> ""r-^Z = ^> 0-^)

The cross (1, 0, 0) (0, 1, 0) will be singular, having ao' deter-

minations.

In general, if we have

F(iX,iX,iX,) = 0, ,l>(^,,X,,X,) = 0,

the line-congruence can be assembled into oo' surfaces withleft, and cc * surfaces with right pai-atactic generators. Suchsurfaces will have Gaussian curvature zero. We shall showalso in Chapter XVI that the lines of such a congruence arenormals to a series of surfaces of Gaussian curvature zero.

* Apparently nothing has ever been published concerning this type ofcongruence. The theorems here given are taken from an unpublished sectionof the author's dissertation, cit.

CHAPTER XI

THE CIECLE AND THE SPHERE

The simplest curvilinear figures in non-euclidean geometryare circles, and it is now time to study their properties*

Definition. The locus of all points of a plane at a constant

distance from a given point which is not on the Absoluteis called a circle. The given point shall be called the centre

of the circle, its absolute polar, which wiU also turn out to

be its polar with regard to the circle, shall be called the axis

of the circle. A line through the centre of the circle shall becalled a dianveter. Let the reader show that all points of

a circle are at constant distances from the axis, a distance

whose measure becomes infinite in the limiting euclidean case.

To get the equation of the circle whose centre is (a) andwhose radius is r, i.e. this shall be the measure of the distance

of all points from the centre, we have

(ax) r^ =COS:--

-/(aa) -/{xx) ^

cos2^(aa) (xx)-{axy = 0. (1)

rIt is evident that when cos^ y ^ 0, this curve has double

contact with the Absolute, the secant of contact being the

axis, and, conversely, every such curve of the second order

will be a circle. The absolute polar of a circle will, hence,

be another circle, so that the circle is self-dual :

—

Theorem 1. Definition. The Theorem 1'. The envelope

locus of all points of a plane of all lines of a plane which

at a constant distance from make a constant angle with

a given point thereof is a a given line is a circle having

circlewhose centre is the given the given line as axis.

point.

Note that a circle of radius -g- is a line, and that circle of

radius is two lines.

* For a very simple treatment of this subject by means of pure Geometry,

see Riccordi, ' I cercoli nella geometria non-euclidea,' Qiomale di Matematica.

zviii, 1880. Eiccordi's results had previously been reached analytically byBattaglini, ' Sul rapporto anarmonico sezionale e tangenziale delle coniche,'

ibid.,zii, 1874.

I2

132 THE CIRCLE AND THE SPHERE CH.

Restricting ourselves, for the moment, to the real domainof the hyperbolic plane, we see that if the centre be ideal,

the axis will be actual, and the curve will appear in the actual

domain as the locus of points at a constant distance from the

axis, an actual line. In this case the circle is sometimes

called an equidistant curve. If the centre be actual we shall

have what may be more properly called a proper circle.

Notice that to a dweller in a small region of the hyperbolic

space, a proper circle would appear much as does a euclidean

circle to a euclidean dweller, wlule an equidistant curve wouldappear like two parallel lines. These distinctions will,

naturally, disappear in the elliptic case ; in the spherical, the

circle wUl have two centres, which are equivalent points.

If the point (a) tend to approach the Absolute (analytically

speaking) the equation (1) will tend to approach an inde-

terminate form. The limiting form for the curve will bea conic having four-point contact with the Absolute. Such acurve shall be called a horocycle, the point of contact beingcalled the centre, and the common tangent the axis. If (u) bethe coordinates of the axis, we have

(v/u) = 0,

and the equation of the horocycle takes the form

(V + ^^2*) (aâ;) + C (ux)* = 0.

Theorem 2. A tangent to Theorem 2'. A point on aa circle is perpendicular to the circle is orthogonal to thediameter through the point of point where the tangent there-

contact, at meets the axis.

These simple theorems may be proved in a variety of ways.For instance every circle will be transformed into itself bya reflection in any diameter, hence the tangent where thediameter meets the curve must be perpendicular to the diameter.

Or, again, if AB = AC,a line from A to one centre of gravityof B, C will be perpendicular to BC; then let B and G closeup on this centre of gravity. Or, lastly, the equation of thetangent to the circle (1) at a point (y) will be

(xy) (oa)— iV(aaj) (ay) = 0.

The diameter through (y) will have the equation

I

xyaI

= 0.

If we indicate these two lines by (u) and (v), then

(uv) = (oa)I

yay \-N{ay) \aya\.

Let the reader show that these theorems hold also in thecase of the horocycle.

XI THE CIRCLE AND THE SPHEKE 133

Theorem 3. The locus ofthe centres of gravity of pairs

of points of a circle whoselines are concurrent on theaxis, is the point of concur-

rence, and the diameter per-

pendicular to these lines.

Theorem 4. If two tangents

to a circle (horocycle) makea constant angle, the locus of

their point of intersection is

a concentric circle (horocycle).

TheoreTTiB'. The envelopeof the bisectors of the angles

of tangents to a circle frompoints of a diameter, is this

diameter, and its absolute

pole.

Theorem 4'. If two points of

a circle (horocycle) are at aconstant aistance,theenvelope

of their line is a coaxal circle

(horocycle).

The element of arc of a circle of radius (r) will be, byChapter IV (5),

rds = ksiardO.

The circumference of tiie circle is thus

A; sin ksink

Let the tangents at P and P' meet at Q, the centre of the

circle being A. Let A<^ be the angle between the tangents,

and let P" be the point on the tangent at P whose distance

from P equals PP , or, in the infinitesimal, equals ds. TheAPAP' and AP'PP" are isosceles, hence

A^ = 24-P'PP",

p'p"4 tan ——r-

hmit -T- = limit ==1ds PP'

PP'sin

'ZP'P'= limit .

PP''

k

ButPQ • »'x ,7. ds

i&^-f = sin-r tanôt^ = ^r ,

tan^ = tan^ cos ^ (w- A«/>) by IV (6),

PQ rlimit tan -j7 = | tan t A ^


Hence

Hmit^ = limit ?££' =^- . (2)

ic

We shall subsequently define this expression as the curva-

ture of the circle at the point (P). We see that, as we should

expect, it is constant.

We shall next take up simple systems of circles. We leave

to the reader the task of making the slight modifications in

what follows necessary to adapt it to the case of spherical

geometry. In the general case two circles, neither of whichis a line, will intersect in four points, real, or imaginary, in

pairs. If two circles lie completely without one another they

will have four real common tangents, the absolute polars of

such circles will interaect in four real points. The difficulty

of visualization disappears in the hyperbolic case where wetake one at least of the circles as an equidistant curve. If weidentify the euclidean hemisphere, where opposite points of

the equator are considered identical, with the elliptic plane,

we see how two circles there also can intersect in four real

points. In the spherical case, by Chapter VHI, the Absolute is

the locus of all points which are identical with their equiva-lents. A point will have one absolute polar, a line twoequivalent absolute poles. The absolute polar of a circle is

two equivalent circles, which are also the absolute polars ofthe equivalent circle. Two real circles cannot intersect inmore than two real points.

Two circles which intersect in four points will have threepairs of common secants. The problem of finding the commonsecants of two conies will, in general, lead to an irreducibleequation of the third degree. When, however, the two conieshave double contact with a third, the equation is reducible,and one pair of secants appears which intersect on the chordsof contact, and are harmonically separated by them.* In thecase of two circles these secants shall be called the radicalaxes. They will

Theorem 5. If two circles Theorem 5'. If two circlesintersect in four points, two have four common tangents,common secants called radical two intersections of these,axes are concuiTent with the called centres of similitude, lie

axes of the circles and har- on the line of centres, are

* This theorem is, of coxirse, well known. Cf. Salmon, Conic Sections,sixth edition. London, 1879, p. 242.

XI THE CIRCLE AND THE SPHERE 135

monically separated by them, harmomcallj separated by theThey are perpendicular to one centres and are mutuallyanother and to the line of cen- orthogonal. The bisectors oftres. The centres of gravity angles of the tangents at aof the intersections of the centre of similitude are thecircles with a radical axis are line of centres and the line to

the intersections with the the other centre of similitude,

other radical axis and withthe line of centres.

If the equations of the two circles be

cos^ ^ (oa) (xx) - (axy = 0, cos^ ^ (bb) (xx) - (bx)^ = 0,

the equations of the radical axes will be

cos -r V{bb) (ax) + cos -^ v{aa) {bx)\

(cos^ 7(66) (ax) - cos j -/(oo) (6a;))= 0, (3)

The last factor equated to zero will give

(ax) (bx)

-/(oa) V{xx) _ '/(bb) -/{xx)^

cos- cosh

and the two sides of this equation will, by Ch. IV (4), be the

cosines of the A;th parts of the distances from (oo) to the points

of contact of tangents, thence to the two circles.

Theorem, 6. Ifa set of circles

through two points have the

line ofthese points as a radical

axis, the points of contact of

tangents to all of them from

a point of the line lie on a

circle whose centre is this

point.

Theorem &. Ifa set of circles

tangent to two given lines

have the intersection of the

lines as a centre of similitude,

the envelope of tangents to

them at the points where they

meet a line through this

centre of similitude will bea circle with this line as axis.

Consider the assemblage of all circles through two given

points. If the line connecting the two points be a radical

axis for two of these circles it will be perpendicular to their

line of centres at one centre of gravity of the two points, and

in every case a perpendicular from the centre of a circle on


a secant will meet it at a centre of gravity of the two points

of the circle on that line. We thus see

—

Theorem 7. The assemblage

of all circles through twocommon points will fall into

two famdlies according as the

perpendicular from the centre

on the line of these points

passes through the one or the

other of their centres of

gravity. Two circles of the

samefamily,andtheyonly,willhave the Ime as a radical axis.

Let us now take a third point, and consider the circles that

pass through all three.

Theorem 8. Four circles will

pass through three given

Theorem 7'. The assemblage

of all circles tangent to twolines will fall into two families

according as the centres lie onthe one or the other bisector

of the angles of the lines.

Two circles of the samefamily, and they only, will

have the intersection of the

lines as a centre of similitude.

points. Each line connecting

two of the given points will

be a radical axis for two pairs

of circles.

Theorem, S'. Four circleswill

touch three given lines. Eachintersection of two lines will

be a centre of similitude for

two pairs of circles.

Tlieorerti 9. The radical axes Theorem, 9'. The centres of

of three circles pass by threes similitude of three circles lie

through four points. by threes on four lines.

Of course when two circles touch one another, their commontangent replaces one radical axis, and the point of contact onecentre of similitude. Two circles will have double contactwhen, and only when, they are concentric. We get at oncefrom (6) and (9)

Theorem 10. Four circles

may be constructed to cuteach of three circles at right

angles twice.

Theorem 10'. Four circles

may be constructed so thatthe points of contact of tan-gents common to them and toeach of three given circles

form two pairs of orthogonalpoints.

It is here assumed that no two of the given circles are con-centric. There is no reason to expect that because two circles

intersect at right angles in two points they will in the othertwo. Let the circles be

cos" 5 {aa) {xx)- {axf= 0, cos"^ {hh) {xx)- {hxY = 0.


Let (y) be a point of intersection ; the lines thence to thecentres ai"e

\xya\ = 0, \xyb\=0.

The cosine of the angle foimed by them will be

coaO =(yy) (ay)

(by) (ab)

Ayy) {^)-(o'yy Ayy) {i>b)- {byf

(a6)— cos-jT COS -^ V{aa) V(bb)

sin -— sin -^ V(aa) V(bb)

(4)

This gives two values for the angle which will be equal when,and only when , ,

.

•^(ab) = 0.

The condition of contact will be

cosfl=+l, cos(^ + ^) =-=^^;and of orthogonal intersection

(5)

cos -î cos -r =^2 _ (®^)

* *; ^/{aa) -^{bb)(6)

these last two facts being, also, geometrically evident. Wesee that two circles cannot have four rectangular intersections,

for if

(a5) = 0, cos^ = 0. C)

the circle is a line.

Theorem 11. The necessary

and sufficient condition that

two circles should cut at the

same angle at all points is

that their centres should bemutually orthogonal.

TheoreTn 11'. The necessary

and sufficient condition that

two circles should determineby their points of contact,

congruent distances on all four

common tangents, is that theiraxes should be mutually per-

pendicular.

Notice that these two conditions are really identical.


We shall define as a sphere that surface which is the locus

of all points of space at congi-uent distances from a point not

on the Absolute.

Tlieorem 12. A sphere is

the locus of all points at a

constant distance from a given

point not on the Absolute.

It is, when not a plane, aquadric with conical contact

with the Absolute.

Theorem 12'. A sphere is

the envelope of planes meeting

at a constant angle a plane

which is not tangent to the

Absolute. It is, when not a

point, a quadric with conical

contact with the Absolute.

Note that a plane and point are special cases of the sphere.

The fixed point shall be called the centre, the plane of conical

contact the axial plane of the sphere. A line connecting anypoint with the centre of a sphere is perpendicular to the polar

plane of the point, a tangent plane is perpendicular to the line

from the point of contact to the centre, to the diameter throughthe point of contact let us say.

Theorem 13. Two spheres

will intersect in two circles

whose planes are perpen-dicular to the line of centres

and to one another, and are

harmonically separated by the

axial planes.

Theorem 14. Three spheres

not containing a commoncircle will meet in three pairs

of circles whose planes are

collinear by threes in fourlines.

TheoreTYi 15. Four sphereswhose centres are not co-

planar intersect in twelvecircles whose planes pass bysixes through eight pointswhich, with the centres of thespheres, form a desmic con-figuration.

Theorem 16. The necessaryand sufficient condition thattwo spheres should cut at thesame angle along their two

Theorem 13'. The commontangent planes to two spheres

envelop two cones of revolu-

tion whose vertices are

mutually orthogonal andhai-monically separated bythe centres.

Theorem 14'. Three spheres

not tangent to a cone of re-

volution have three suchpairs ofcommon tangent coneswhose vertices are collinear in

thi'ees on four lines.

Theorem 15'. Four sphereswhose axial planes are notconcurrent are enveloped in

pairs by twelve cones of re-

volution whose vertices lie

by sixes in eight planes which,with the axial planes, deter-

mine a desmic configuration.

Theorem 16'. The necessaryand sufficient condition that

two spheres should, by their

contact, determine congruent


circles is that their centres distances on the generators ofshould be mutually ortho- the two circumscribed cones,gonal. is that their axial planes

should be mutually perpen-dicular.

We shall terminate this chapter by giving an unusuallyelegant transformation from euclidean to non-euclidean space.*

Let us assume that we have a euclidean space where a pointhas the homogeneous coordinates x, y, z, t and a hyperbolicspace for which k^ = — 1, a point being given by our usual (x)

coordinates. Let us then write

px = x^, py= X2, pz= -/x^-Xj^—x^^-x./, pt = Xf,-x.. (8)

To each point of hyperbolic space will correspond twopoints of euclidean space. Let us choose that for which the

real part of Vx^^—x^—x^—x^ is greater than zero. Whenthe real part vanishes, we may, by adjoining to our domain ofrationality a square root of minus one, distinguish between the

imaginary roots, and so choose one in particular. We maythus say that to every point of hyperbolic space, not on the

Absolute, will correspond a point of euclidean space abovethe plane z = 0, and to each points of the Absolute will

correspond points of this plane. The transformation is real,

so that real and actual points will correspond to real ones.

Converaely, we get from (8)

ax^z=x^-^'i^-\-z^+t^, <iXi = 2oct, a-x.^= 2yt,

ai!3= a;2 + 2/^ + ;2-n (9)

and to each point of euclidean space, above, or on the z plane,

will correspond a point of hyperbolic space, not on, or on the

Absolute.

Suppose that we have a euclidean sphere of centre (a, 6, c, d)

and radius r. If we write for short

(a2 + 62 + c2_dV)=;A

the equation of this sphere may be written

{dx-atY + {dy-bty+ {dz-ety = d^rH^

d^{a^ + y^ + z'')-2dt{ax+ lnj + cz)+pH^ = 0. (10)

* This transfoTmation seems to have been fiist given in the second edition

of Wissensehaft und Hypothese, by Poincare, translated by F. and L. Lindemann,Leipzig, 1906, p. 258. This is fruitfully used in the dissertation of Miinich,' Kicht-euJclidische Cykliden,' Munich, 1906. We have adapted the notation

to conform to our own usage.

140 THE CIRCLE AND THE SPHERE ch.

Transforming we get, after splitting oflF a factor x^—x^ which

con-esponds to the euclidean plane at infinity,

cP (io + aJj)- 2d {axi + bx^+ c Vx^^-x^- x^- x^)

[(dHi>-)i;o-2ada;,-2Ma;2 + (d2-/)a;3]2

= ^c^d;'(x^--x^--xi-x^). (11)

This is a sphere of hyperbolic space whose centre is

and whose radius r, is given by

cosh /•, = =^ •

Convereely, if we have the hyperbolic sphere

(«0«^0- I'h*!- «2«'l2—^sS's)^

= cosher, (V-ai^-d;.''- dj^) (x^-x^-xi-xi), (12)

we get from (9)

[(do- «3) (a;' + 2/' + s')- 2dia!f-2d,yf + (do + d3)t']

= +2cosh7'i Vd/— dj'*—dj*— dg^zf. (13)

We have here two spheres which differ merely in the z coor-

dinate of their centre, i.e. two spheres which aie reflections

of one another in the z plane. If the hyperbolic sphere werereal and actual, one of the euclidean spheres would lie whollyabove the s plane, and the other wholly below it. We maysay that (leaving aside special cases) a hyperbolic sphere will

correspond to so much of a euclidean sphere as is above or

in the z plane, and to the reflection in the z plane of so muchof the sphere as is below it.

A euclidean sphere for which c = 0, that is, one whosecentre is in the z plane will correspond to a plane in hyperbolic

space, a hyperbolic sphere for which

do-d3=0,

that is, one whose centre is in the plane which corresponds

to the euclidean plane at infinity, will correspond to a planein euclidean space. A euclidean circle perpendicular to the

z plane will correspond to a hyperbolic line, a hyperbolic circle

which is perpendicular to the plane dj—d3=0, will correspondto a euclidean line.

We may go a step further in this direction. Suppose that

we have two euclidean spheres given by an equation of the


type (13), and the condition that they shall be mutually ortho-gonal is that

— djdo' + Aid/ + djd/

± cosh ri cosh r/ Vd^^^^V^^dT^^d/ Va^^-a^^-a^^-a.^^+ aû^ = 0,

cosh rj cosh r/— dfldp^ + d^d/ + d2d/ + aâ^= +

-Z^dT+dT+dT+dJ^ A/-d„"'+di'« + d2'^+d„'='

But this gives immediately that the corresponding hjrperbolic

spheres are also mutually orthogonal, and conversely. Wethus have a correspondence of orthogonal spheres to orthogonalspheres. We see next that the lines of ciu'vature of anysurface will go into any lines of curvature of the correspondingsurface, and hence the Darboux-Dupin theorem must hold in

hyperbolic space, namely, in any triply orthogonal systemof surfaces, the intersections are Imes of curvature.

Were we willing to sacrifice the real domain, we might in

a similar manner establish a con'espondence between spheres

of euclidean and of elliptic space.

CHAPTER XII

CONIC SECTIONS

The study of the metrical properties of conies in the non-

euclidean plane, is, in the last analysis, nothing more nor less

than a study of the invaiiants and covariants of two conies.

We shall not, however, go into genei"al questions of invariant

theory here, but rather try to pick out those metrical pro-

perties of non-euclidean conies which bear the closest analogy

to the corresponding euclidean properties.*

First of all, let us classify our conies under the real con-

gruent group ; that is, in relation to their intersections with

the Absolute. This may be done analytically by means of

Weierstrass's elementary divisors, but the geometric question

is so easy that we give the results merely. We shall begin

with the real conies in the actual domain of hyperbolic space.

(1) Convex hyperbolas. Four real absolute points, no real

absolute tangents.

(2) Concave hyperbolas. Four real absolute points, four

real absolute tangents.

(3) Semi-hyperbolas. Two real and two imaginary absolute

points and tangents.

(4) Ellipses. Four imaginary absolute points and tangents.

(5) Concave hyperbolic parabolas. Two coincident, andtwo real and distinct absolute points and tangents.

(6) Convex hyperbolic parabolas. Two coincident, and tworeal and distinct absolute points. Two coincident, and twoconjugate imaginary absolute tangents.

(7) Elliptic parabolas. Two coincident, and two conjugate

imaginary absolute points and tangents.

(8) Osculating parabolas. Three real coincident, and onereal distinct absolute point, and the same for absolute tangents.

• The treatment of conies in the present chapter is in close accord withthree articles by D'Ovidio, 'Le propriety focali delle coniche,' 'Sulle conicheconfocali,' and ' Teoremi sulle coniche ', all in the Atti ddla R. Atxademia delle

Saenze di Torino, vol. xxvi, 1891. These articles suffer from the curiousblemish, not uncommon in Italian mathematical publications, that thetheorems are not given in distinctive type. See also Story, ' On the non-euclidean Properties of Conies,' American Journal of Mathematics, vol. v, 1882 ;

Killing, ' Die nicht-euklidische Geometrie in analytischer Behandlung,'Leipzig, 1885, and Liebmann, • Nicht-euklidische Geometric,' in the SamnUungSchubert, zliz, Leipzig, 1904.

CH. XII CONIC SECTIONS 143

(9) Equidistant cui-ves.

(10) Proper circles.

(11) Horocycles.

In the real elliptic, or spherical, plane, we shall havemerely

—

(1) Ellipses;

(2) Circles.

In what follows we shall limit ourselves to central conies,

i.e. to those which cut the Absolute in four distinct points.

A real central conic in the actual domain of the hyperbolicplane will have a common self-conjugate triangle with the

Absolute which is real, except in the case of the semi-hyper-bola. In the elliptic case it will surely be real. Taking this

as the coordinate triangle we may write the equation of theAbsolute in typical form, while that of the conic is

"^CiX^ = 0. (1)

i

We assume that no two of our c's are equal, and that noneof them are equal to zero.

Our plane being ^^= 0, we shall use the letters h, k, I as

a circular permutation of the numbers 0, 1, 2, and define the

vertices of the common self-conjugate triangles as centres of

the conic, while its sides are called the axes. Be it noticed

that in speaking of triangle in this sense we are using the

terminology of projective geometry where a triangle is a figure

of three coplanar, but not concurrent lines, and not the exact

definition of Chapter I, which is meaningless except in a re-

stricted domain. There will, however, arise no confusion

&om this.

Theorem 1. Each centre of Theorem, V. Each axis of

a central conic is a centre a central conic is a bisector

of gravity for every pair of of an angle of each pair of

points of the conic coUinear tangents to the conic con-

therewith, current thereon.

The three pairs of lines which connect the pairs of intersec-

tions of a central conic with the Absolute shall be called

its pairs of focal lines. The three pairs of intersections

of its absolute tangents shall be called its pairs of foci.

[ Theorem 2. Conjugate points Theorem, 2'. Conj ugate lines

of a focal line of a conic are through a focus of a conic are

mutually orthogonal. mutually perpendicular.

144 CONIC SECTIONS ch.

Theorem 3. Two focal lines Theorem 3'. Two foci of a

of a central conic pass thi-ough central quadxic lie on each

each vertex, and are perpen- axis, and are orthogonal to

dicular to the opposite axis. the opposite centre.

The coordinates of the focal lines Z^./^', through the centre

Ui = 0, will be

Ufc : Ufc :uj = : Vc^^j, : ± Vci-c^^. (2)

The coordinates of the foci F^, F^ on the opposite axis

'^^x^:x„:xi = 0: v^c, {c^-c^) : ± v^c, (ci-c,). (3)

The polars of the foci with regard to the conic shall be called

directrices, the poles of the focal lines its director pointa.

A directrix cZj pei"pendicular to the axis ic^ will have the

"1"**^°^v^;J^'^)x,+ ^/^;^^^:^)Xl = o. (4)

Let {x) be a point of the conic. Eliminating x/^ by means

of (1) we get( ^^^ ^ ^^

(XX) = ^^^h — — ^331^

We then have

^P-Ffc ^ "/C; (Cft-Cfc) Xj, + '/Cfe (C; -C;,) Xj

sin^ = '^''kich-Ck)^k+ ^ci(ci-c^)xi^g^

*= -/(cj- Cfc) ajfc"- (cj- cj) xj" y(Cfc

- ci)

If (2;^ be the corresponding directrix

SlUj^

— y —

.

_

the signs of the radicals in the numerators of the two ex-pressions being the same

sin—r-^ /

—

:: ;

(7)

"°^Pd,^ ^/c,,{Cj,-Ci)

= /'^^^- («)

xit CONIC SECTIONS 145

Theorem 4. The ratio of the Theorem 4'. The ratio ofsines of the kih parts of the the sines of the angles whichdistances from a point of a a tangent to a central coniccentral conic to a focus and makes with a focal line andto the corresponding directrix the absolute polar of theis constant. corresponding director point

is constant.

tan^ 1Ml tanH^^ tanH^= tan''^ÂA'tanH^/fc/fc'tanH^/,//= 1. (9)

sin?^ sin^^' = "fc(''^-^fc)^fc'-îfci-''fe)'"t'

k k [{cicj) xf- (Cft- Ck) aJfc^] {cj,-ci)

Cfe(Cit-Cj)(a;a;)'

. PF. . PFj/ . PFj, . PFrf . PFi . PF{sm -j^ sm—^ : sin —r^ sin —r^ : sm —r^ sm —^-i-

K /C K K tC IC

Ca ("k - cj) Cfc (cj -c,y Ci (C,,- Cj.)(10)

PF. PF.' PFj, PFj! PFi PF,' „ ,„,CSC —T^ esc -r^ + CSC -r^ CSC —r^ + CSC -~ esc -r-^ = 0. (11)

tC iC IC tC iC iC

,-?^*cos-^'= <=l(Oh-Ck)'^k'-Ckici-''h)î'

I k - k j cj,-ci^

cosI _

, .rPjPfe P^hl, iT-PÂ- -P-^fc'1. iF-P^z -f^j'1

= ±1. (13)

With regard to the ambiguity of signs : the upper sign in

(12) will go with the upper sign throughout in (13), and so

for the lower sign. It is a£o geometrically evident that

in the case of an ellipse we must take the upper, and in

the case of a hyperbola the lower sign (when in the real

domain).COOLlDfiS K

146 CONIC SECTIONS CH.

Theorem 5. The sum of the

distances from real points of

an ellipse and the difference

of the distances from real

points of a hyperbola or semi-

hyperbola to two real foci onthe same axis is constant.

Theorem 5'. The sum of the

angles which the real tangents

to an ellipse or convex hyper-

bola, or the difference of the

angles which the real tangents

to a concave hyperbola or a

semi-hyperbola make with

two real focal lines through

a centre is constant.

Reverting to our point (x) we see

sin4^ = ^Cfe-<'fca'fe+ A-gftî

V c,,

sm—r^sin-V^ = H 2_k k ~ ('k~'^l

Theorem 6. The product ofthe sines of the A;th parts

of the distances from a pointof a central conic to two focal

lines through the same centreis constant.

Theorem 6'. The product of

the sines of the ^th parts

of the distances to a tangentfrom two foci of a central

conic on the same axis

constant.

IS

Let us now recall Desargues' theorem, whereby a transversal

meets the conies of a pencil in pairs of points of an involution.This will apply to a central conic, the Absolute, and the pairs

of focal lines. A dual theorem will of course hold for a centralconic, the Absolute, and the pairs of foci.

Theorem 7. The intersec-

tions of a line with a central

conic, and with its pairs ofcorresponding focal lines, all

have the same centres ofgravity.

Theorem, 8. The polar of apoint with i-egard to a central

conic passes through onecentre of gravity of the inter-

sections of each focal line withthe tangents from the point to

the conic.

Theorem T. The tangentsfrom a point to a central

conic, and the pairs of lines

thence to its paii-s of corre-

sponding foci, form angleswith the same two bisectors.

Thewem 8'. The pole of aline with regard to a central

conic lies on one bisector of

the angle determined at eachfocus by the lines thence to

the intersections of the givenline with the conic.

XII CONIC SECTIONS 147

A variable point of a conic ^111 determine projective pencilsat any two fixed points thereof, and these will meet any linein projective ranges, hence

Theorem 9. If a variablepoint of a central conic beconnected with two fixedpoints thereof, the distancewhich these lines cut on anyfocal line is constant.

Theorem 9'. If a variable

tangent to a central conic bebrought to intersect two fixed

tangents thereof, the angle of

the lines from a chosen focus

to the two intersections is

constant.

Becalling the properties of the eleven-point conic of twogiven conies and a line

:

Theorem 10. If a line anda central conic be given, the

two mutually conjugate andorthogonal points of the line,

the points of the focal lines

orthogonal to their inter-

sections with the line, and the

three centres lie on a conic.

Theorem, 10'. If a point anda central conic be given, thetwo lines through the point

which are mutually conjugate

and perpendicular, the perpen-diculars on the line from thefoci, and the three axes all

touch a conic.

It is a well-known theorem that the locus of points, whencetangents to two conies form a harmonic set, is a conic passingthrough the points of contact with the common tangents.


points whence tangents to acentral conic are mutuallyperpendicular is a conic meet-

ing the given conic where it

meets its directrices.

Theorem, 11', The envelopeof lines which meet a central

conic in pairs of mutuallyorthogonal points is a conic

touching the tangents to thegiven circle from its director

points.

It is clear that neither of these conies wiU, in general, be

a circle, as in the euclidean case. If the mutually perpen-

dicular tangents from the point {y) be

{ux) = 0, {vx) = 0.

0..2 2 «"2,,.2 0-2

''U.!'0..2 2

2 ^ (w)+ ^l{uu)-2^ ^»(u.) = 0,

k2


'^c^{ck + ci)yi' = 0. (14)

h

Let the reader show that the equation of the other conic

will be 2

2(Cfc+cjX = o.

h

We may extend the usual euclidean proof to the first of the

following theorems

—


the reflection of a real focus

of an ellipse in a variable

tangent, is a circle whosecentre is the corresponding

focus.

Theorem 12'. The envelopeof the reflection in a variable

point of an ellipse, of a real

focal line, is a circle whoseaxis is the corresponding focal

line.

Let (y) be the coordinates of a point P of our conic. Theequation of a line through the centre 0,^ conjugate to the line

0,,-Pwmbe c^yj,xj, + ciyixi = 0.

This will meet the conic in two points P' having the coor-dinates /

tan^^+tan^gg^=-^^fa + ^').(15)

Theorem 13. The sum of

the squares of the tangentsof the kth parts of the dis-

tances from a centre of acentral conic to any pair ofintersections with two con-jugate lines through this

centre is constant.

Theorem 13'. The sum ofthe squares of the tangentsof the angles which an axis ofa central conic makes with apair of tangents to the curvefrom two conjugate points ofthis axis is constant.

We shall call two such diameters as OjP, O^P" conjugatediameters.

sin ÔjP'= (Ml+Mi')yf

tanO.P. O^P'

-^yk^+yi^ ^cj.^vic'+ci^yr

A_tan _^sin^PO^P'^ +


Theorem, 14. The productof the tangents of the A;th

parts of the distances from acentre of a central conic to

two intersections with a pairof conjugate diametersthroughthat centre, multiplied by thesine of the angle of these

diameters is constant.

Theorem, 14'. The productof the tangents of the angles

which an axis of a central

conic makes with two tangents

to it from a pair of conjugate

points of this axis, multiplied

by the sine of the Mix pajt of

the distance of these points is

constant.

The equation of a line through the centre Oj perpendicular

to OjP will be yj,x„ + yixi = 0.

This will meet the conic in points P" having coordinates

-VXj,:xj,:xi

op"_ -y-jom'+c^yi')

Vi -Vk'

cos-

îch-ci)yu'+(<ih-0k)yi'

^^^.0P"_ -icm'+c^yi')k ChiVh+yi^)

ctn^-y +ctn^—t;- (16)

Theorem 15. The sum of

the squares of the cotangents

of the fcth parts of the dis-

tances from a centre of a

central conic to two inter-

sections of the curve with

mutually perpendicular dia-

meters through this centre is

constant.

Cfc + Cj.

Them'eTn 15'. The sum of

the squares of the cotangents

of the angles which an axis of

a central conic makes withtwo tangents from a pair of

orthogonal points of this axis

is constant.

The equation of the tangent f at the point P' is

Chyh'^h+ '<î(^hyi-'»iyk) = o-

From this we get

sin'Oft*'. ('hW

tan

{<>l-ci)cj,yk^+ (c^-c^)ciyi«'

tan -2- =k

'^<^kH

(17)


Theorem 16. The product

of the tangents of the kih.

parts of the distances froma centre of a central conic

to a point of the curve andto the tangent where the curve

meets a diameter conjugate to

that from the centre to the

point of the curve, is con-

stant.

Theorem 16'. The product

of the tangents of the angles

which an axis of a central

conic makes with a tangent

and with the absolute polar

of a point of contact with

a tangent from a point of this

axis conjugate to the inter-

section with the given tan-

gent, is constant.

The equations of two conjugate diameters through 0^ havealready been wi'itten

yi^^k-Vkî = 0. Cfci/jiCfc + CiyiXi = 0.

The product of the tangents of the angles which they makewith the xj, axis is

y^^^yi _ t-j

yi<^kyk

Theorem 17. The product

of the tangents of the angles

which two conjugate dia-

meters through a centre makewith either axis through this

centre is constant.

Theorem 17'. The productof the tangents of the kthparts of the distances of twoconjugate points of an axis

from either centre on this

axis is constant.

Let P;, , Pft' be the intersections of the XJ^ axis with the conic

cosPhPh Cft + Cj

p p ' p p ' p p '

tan2i£ftp..tanH^^-^-tan4i^= -1. (18)

Theorem 18. The productof the squares of the tangentsof the 2^th parts of thedistances determined by acentral conic on the axes is

equal to —1.

Theorem 18'. The productof the squares of the tan-gents of the half-angles of thepairs of tangents to a centralconic from its centres is con-stant.

If a circle have double contact with a conic, we have, withthe Absolute, the figure of two conies having double contactwith a third, already studied in the last chapter.

Theorem, 19. If a circle havedouble contact with a conic,

its axis and the lines connect-

Theorem 19'. Ifa circle havedouble contact with a conic,

its centre and the intersections

xii CONIC SECTIONS 151

ing the points of contact are of the common tangents areharmonically separated by a harmonically separated by apair of focal lines. pair of foci.

Of course we mean by foci and focal lines of any conic whatwe mean in the special case of the central conic.

A circle which has double contact with a central conic

where the latter meets an axis is called an auxiliary circle.

There will clearly be six such circles, their centres being thecentres of the conic. Consider the circle having its centre

at Oji while it has double contact with our central conic at

the intersections with X}^ = 0.

0..2

i

0..2

2 CiXi^ + {cicj) xj,^ = CftV + <^h<^k + '^lî' = 0-

i

This will meet the line (u) through 0^ in points Q, Q', having

coordinates

The same line wiU meet the conic in points P, P', having

coordinates

-'-.=V-"-*^CiUrT+ CiMf

tan'a^^-ci{ul +u^^ ^^^,OftP^-Cfe

tanM:tan^=./q:^/^. (19)

Let us remark, finally, that the tangent of the fcth part of the

distance from a point to a line, is the cotangent of the Ath part

of its distance to the pole of tlie line, and that if the tangents

of two distances bear a constant ratio, so do their cotangents :

Theorem 30. If the tangents Theorem 20'. If the tangents

of the Jkth parts of the dis- of the angles which the tan-

tances from the points of a gents to a circle make with a

circle to any diameter be diameter be altered in a con-

152 CONIC SECTIONS ch.

altered in a constant ratio, the stant ratio, the envelope of the

locus of the resulting points resulting lines will be a conic

will be a conic having the having the given circle as an

given circle as an auxiliary. auxiliary circle.

The normal at any point of a conic is the line connecting

it with the absolute pole of its tangent. This line is also

perpendicular to the absolute polar of the given point, so that

the conic and its absolute polar conic are geodesically parallel

curves. The equation of the normal to our conic (1) will be

2'-^'«'* = o- (20)

The tangents to a central conic from a centre shall be called

asymptotes. The equation of the pair of asymptotes through

the centre (0^) will evidently be

CfcXfc* + cja;j^ = 0. (21)

The tangent at the point P with coordinates (y) will meetthem in two points R, R', whose coordinates are

xj,:x,,:xi= _ _^-<^k'-'e('^-'-'iyi± ^''kyk) +('hyh^-<'i -"kVh -^Ck,

t^0^tan^'=: (^^-^^yy^' -eîg^r^). (22)k k c^ci{c],yj,^ + ciyi'^) c^ci

Theorem 21. The product of Theorem 21'. The productthe tangents of the ^th parts of the tangents of the angles

of the distances from a centre which an axis of a central

of a central conic to the in- conic makes with the lines

tersection with the asymptotes from a point of the curve tothrough that centre of a tan- the intersections of the curvegent is constant. with this axis is constant.

A set of conies which meet the Absolute in the same fourpoints shall be said to be homothetic. IS they have the samelour absolute tangents they shall be called confocal. We getat once from Desaigues' involution theorem :

—

Theorem 22. One conic TAeorem 22'. One conic con-homothetic to a given conic focal with a given conic will

will pÂs through every point touch every line, and twoof space, and two will touch will pass through every pointevery line, not through a point not on the common tangents


common to all the conies, in to all. The tangents to thesethe centres of gravity of all two will bisect the angles ofpairs of intersections of the thepairsof tangents &om thathomothetic conies with this point to all of the confocalline. conies.

Concentric circles are a special case both of homothetic andof confocal conies. The general form for the equations of conies

homothetic and confocal respectively to our conic (1) will be

2(Ci+m)V = 0. (23); 2,^x^=0. (24)

It is sometimes useful to modify the second of these

equations, in order to introduce the elliptic coordinates of

a point, i.e. the two parameters giving the conies of the

confocal system which pass through it. Let us write - in

place of q.

V(asB)= X,.

Our confocal conies have, then, the general equation

C..2 XT

(25)

If Aj and X^ he the parameter values of the conic through (X)we have , —

X. = /(Cfc-Ct)(Cft-î)(c&-^2)

i 4

'2,''h^('^h-ci)

0..2 0..2

K)dK^''

ii<«~^) n(«~^2)

(26)

(27)

With the aid of these coordinates, we may easily prove for

the non-euclidean case Graves' theorem, namely, if a loop

of thread be cast about an extremely thin elliptic disk, andpuUed taut at a point, that point will trace a confocal ellipse.

We shall not give the details here, however, for in the nextchapter we shall work at length the more interesting corre-

sponding problem in three dimensions, and the calculations

ai'e too &tiguing to make it advisable to carry them through

twice.

CHAPTER XIII

QUADRIC SURFACES

The discussion of non-euclidean quadric surfaces may be

carried on in the same spirit as that of conic sections in the

preceding chapter. There is not, however, the same wealth

of easy and interesting theorems, owing to the greater com-plication in the formation of the simultaneous covariants

of two quadrics.

Let us begin by classifying non-euclidean quadrics underthe group of real congruent transformations.* We beginin the actual domain of hyperbolic space, giving only those

surfaces which have a real part in that domain and a non-vanishing discriminant. The names adopted are intended to

give a certain idea of the shape of the surface. We shall

mean by curve, the curve of intersection of the surface andAbsolute, while developable is the developable of commontangent planes.

A. Central Quadrics.

(1) Ellipsoid. Imaginary quartic curve and developable.

(2) Concave, non-ruled hyperboloid. Real quartic curveand developable.

(3) Convex non-ruled hyperboloid. Real quartic curve,imaginary developable.

(4) Two-sheeted ruled hyperboloid. Real quartic curveand developable.

(5) One-sheeted ruled hyperboloid. Real quartic curve,imaginary developable.

(6) Non-ruled semi-hyperboloid. Real quartic curve anddevelopable.

(7) Ruled semi-hyperboloid. Real quartic curve and de-velopable.

The last two surfaces differ from the preceding ones in that

* The clasBification here given is that which appears in the author'sarticle 'Quadric Surfaces in Hyperbolic Space', Transactions of the AmericanMathetnatical Society, vol. iv, 1903. This classification was simplified andput into better shape by Bromwich, ' The Classification of Quadratic Loci,'ibid., vol. vi, 1905. The latter, however, makes use of WeierstrassianElementary Divisors, and it seemed wiser to avoid the introduction of theseinto the present work. Both Professor Biomwich and the author wrote inignorance of the fact that they had been preceded by rather u crude articleby Barbarin, ' Etude de gôm^trie non-euclidienne,' Uemoires amrannis parVAcademic de Belgigfue, vol. vi, 1900.

CH. XIII QUADRIC SURFACES 155

here two vertices of the common self-conjugate tetrahedron(in the sense of projective geometry) of the surface andAbsolute are conjugate imagiuaries, while in the first fivecases all four are real.

B.

(8) Elliptic paraboloid. Imaginaa-y quartic curve with realacnode, imaginary developable.

(9) Tubulai" non-ruled hyperbolic paraboloid. Real quarticwith acnode, real developable.

(10) Cup-shaped non-ruled hyperbolic paraboloid. Realquartic with acnode, imaginary developable.

(11) Open ruled hyperbolic paraboloid. Real acnodalquartic, real developable.

(12) Gathered ruled hyperbolic paraboloid. Real crunodalquartic, imaginaxy developable.

(13) Cuspidal non-ruled hyperbolic paraboloid. Real cus-pidal quartic curve, real developable.

(14) Cuspidal ruled hyperbolic paraboloid. Real cuspidalquartic curve, real developable.

(15) Horocyclic non-ruled hyperbolic paraboloid. The curveis two mutually tangent conies, developable real.

(16) Horocyclic eUiptie paraboloid. Curve is two mutuallytangent imaginary conies, developable imaginary.

(17) Horocyclic ruled hyperbolic paraboloid. Curve is tworeal mutually tangent conies, developable imaginary.

(18) Non-ruled osculating semi-hyperbolic paraboloid. Thecurve is a real conic and two conjugate imaginary generators

meeting on it. The developable is a real cone, and twoimaginary lines.

C. Surfaces of Revolution.

(19) Prolate spheroid. Curve is two imaginary conies in

real ultra-infinite planes, imaginary developable.

(20) Oblate spheroid. Curve is two imaginary conies in

conjugate imaginary planes meeting in an ultra-infinite line,

imaginary developable.

(21) Concave non-ruled hyperboloid of revolution. Curveis two real conies whose planes meet in an ideal line, real

developable.

(22) Convex non-ruled hyperboloid of revolution. Absolute

curve two real conies whose planes meet in an ideal line,

imaginary developable.

(23) Ruled hyperboloid of revolution. Curve two real

conies whose plês meet in an ideal line, imaginary de-

velopable.

156 QUADRIC SURFACES ch.

(24) Semi-hyperboloid of revolution. The curve is a real

conic, and an imaginary one in a real plane, the developable

is a real cone and an imaginary one.

(25) Elliptic paraboloid of revolution. The absolute curve

is an imaginary conic in an ultra-infinite plane, and twoimaginary generators not intersecting on the conic. The de-

velopable is an imaginary cone, and the same two generators.

(26) Tubular semi-hyperbolic paraboloid of revolution.

The curve is a real conic and two imaginary generators not

intersecting on it ; the developable is the same two lines anda real cone.

(27) Cup-shaped semi-hyperbolic paraboloid of revolution.

Real conic and two imaginary lines not meeting on it. Develop-able same two lines and imaginary cone.

(28) Clifford surface. Curve and developable two generatorsof each set.

D. Canal Surfaces.*

(29) Elliptic canal surface. Curve is two imaginary conieswhose planes meet in an actual line, developable imaginary.

(30) Non-ruled hyperbolic canal surface. Two real conieswhose planes meet in an actual line, developable two real

cones.

(31) Ruled hyperbolic canal surface. Curve two real conieswhose planes meet in an actual line, imaginary developable.

E. Spheres.

(32) Proper sphere. Curve is two coincident imaginaryconies, developable imaginary.

(33) Equidistant surface. Curve two real coincident conies,developable two real coincident cones.

(34) Horocyclic surface. Curve and developable two con-jugate imaginary intersecting generators, each counted twice.

In elliptic or spherical space the number of real varietieswill, of course, be much smaller. We have

(1) Non-ruled ellipsoid.

(2) Ruled ellipsoid.

(3) Prolate spheroid.

(4) Oblate spheroid.

(5) Ruled ellipsoid of revolution.

(6) CliflFord surface.

(7) Sphere.

' Called Stafaces <if Translation in the author's article ' Quadric Surfaces ',

loc. cit.

XIII QUADRIC SURFACES 157

It is worth mentioning that the Clifford surface of elliptic

space has real linear generators, while that in hyperbolic spacehas not.

Let us next turn our attention to that class of quadricswhich we have termed central, and which ai-e distinguishedby the existence of a non-degenerate tetrahedron (in theprojective sense) self-conjugate with regard both to the surface

and the Absolute. The vertices of this tetrahedron shall becalled the centres of the surface, and its planes the axial planes.

When this tetrahedron is chosen as the basis of the coordinatesystem, the Absolute may be written in the typical formwhile the equation of the surface involves none but squaredterms.

Theorem 1. A centre of a Theorem V. An axial planecentral quadric is equidistant of a central quadric bisects

from the intersections with a dihedral angle of every twothe surface of every line tangent planes to the aui-face

through this centre. which meet in a line of this

axial plane.

We obtain a good deal of information about our central

quadrics by enumerating the Cayleyan characteristics of their

curves of intersection with the Absolute, and the con'espondingdevelopables. The curve is a twisted quartic of deficiency

one. Its osculating developable is of order eight and class

twelve. It has sixteen stationary tangent planes, thirty-eight

lines in every plane lie in two osculating planes, two secants,

i.e. two lines meeting the curve twice, pass every point not onthe curve, sixteen poinis in every plane are the intersection of

two tangents, eight double tangent planes pass through every

point. The developable will, of course, possess the dual

characteristics.

Theorem 2. Through an Theorem 2'. In an arbitrary

arbitrary point in space will plane there will be twelve

pass twelve planes cutting a points, vertices of cones cir-

central quadric in osculating cumscribed to a central

parabolas, eight planes of quadric which have stationary

parabolic section will pass contact with the cone of tan-

through an arbitrary line. An gents to the Absolute, eight

arbitrary point will be the points on an arbitrary Unecentre of one section. Sixteen are vertices of circumscribed

planes cut the surface in horo- cones which touch the Abso-

cycles, sixteen points in an lute. An arbitrary plane will

158 QUADEIC SURFACES ch.

arbitrary plane are the centres be a plane of symmetry for

of circular sections, eight one circumscribed cone. Six-

planes of circular section pass teen points are vertices of

through an arbitrary point. circumscribed cones whichhave four-plane contact withthe Absolute. Sixteen planes

through an arbitrary point are

perpendicular to the axes of

revolution of circumscribed

cones of revolution.

The planes of circular section ai'e those which touch the

cones whose vertices are the centres of the quadric, and whichpass through the Absolute curve. It may be shown that not

more than six real planes of circular section will pass throughan actual point, and that only two of these will cut the surface

in proper circles.*

Let us write as the equation of a typical quadric

^CiXi^ = 0. (1)

No two of the c's shall be equal, and none shall equalzero.

The cones whose vertices are the centres and which passthrough the Absolute curves shall be called the focal cones.

In like manner there will be four focal conies in the axial

planes. The equation of the focal cone whose vertex is 0,,

will be

'^(Ci-cnW = 0. (2)

t

The focal conic in the corresponding axial plane will be

^* = 0. 2t=?"'/ = 0- (3)

Let the reader show that each of these conies passes throughtwo foci of each other one.

We next seek the locus of points whence three mutuallytangent planes may be drawn to the surface. Let these be theplanes (v), {w), (oj), and let the equation of the surface and

* See the author's ' Quadric Surfaces ', loc. cit., p. 164.

xiii QUADRIC SURFACES 159

the Absolute in plane coordinates be, in the Clebseh-Aronholdnotation

Uy' = Q, V =V = 0,

Vy'^=U

160 QUADRIC SURFACES CH.

any other such set, and apply Theorem 15 of the samechapter.


squares of the tangents of the

kth parts of the distances

from a centre of a central

quadric to three intersections

of the surface with three con-

jugate diameters through that

centre is constant.


squares of the cotangents of

the kth parts of the distances

from a centre of a central

quadric to three intersections

with the surface of three

mutually perpendicular lines

through that centre is con-

stant.


squares of the tangents of the

angles which an axial plane

of a central quadric makeswith three tangent planes

through three conjugate lines

in that axial plane is con-

stant.


squares of the cotangents of

the angles which an axial

plane of a central quadricmakes with three tangentplanes through three mutuallyperpendicular lines in that

axial plane is constant.

To find the values of the constants referred to in Theorems 4and 5, we have but to choose a pai-ticular set of diameters,say the intersections of the axial planes through 0^. Wethus get

tan^ —^ + tan^ —^— + tan^ —

^

k k k - -< - i * i> <^*

.OkQctn^^+ctn''k

^'+ctn^w:k k

(Cfc + Cj +CjC)

A set of quadrics having the same absolute focal curve, and,hence, the same focal cones, shall be called homothetic. ' A set

inscribed in the same absolute developable, and possessing,in consequence the same focal conies shall be called confocal.

Tltem-em 6. An arbitraryline will meet a set of con-focal quadrics in pairs of

points with the same centres

of gravity.

Theorem 6'. The tangentplanes to a set of confocalquadrics through an arbitraryline, form dihedral angles withthe same bisectors.

Tlieorem 7. Three homo- Theorem, 7'. Three confocalthetic quadrics will touch an quadrics will pass through anarbitrary plane in three arbitrary point, and intersectmutually orthogonal points. orthogonally.

xm QUADRIC SURFACES 161

Let us now set up our system of elliptic coordinates as wedid in the plane

X, = -^, (XX) = I. (8)V{xx)

These coordinates (X) are inapplicable to points of theAbsolute ; we imaê that all such points ai-e excluded fromconsideration. The general equation for the system of quadrics

confocal with that given by (1) will be,* if we replace cj by - .

If the roots be X^, Xj' ^39 ^^ have

V _ / (Cfe-Ai)(Cft-Aa)(Cfe-X3)

''~V(Ok-<'n){Ck-Ci){c,-cJ-^'"^

'e

IF- (SSP(^^^^>- <î>

We wish to express this in terms of our elliptic coordinates.

It will be found that the coefficients of <2X- dK^ will vanish,

and, indeed, this is a priori evident if we have in mind that

our coordinate system is a triply orthogonal one, and the

general formulae for orthogonal curves, as will be shown in

Chapter XY, are the same for euclidean as for non-euclidean

space. We thus get

For the differential of distance we have

dfi* (axe) (dxdx)— (xdx)''

*' *kp ("A-^'i) (''A-''') (Â-"™) ("A-

V

If we give to CJ^ each of its four values, divide the teims into

partial fractions and recombine, we get

ds^ _ 1 1^« (\p-\)(Xp-K)d\p' (.o)

i

The analogy to the corresponding formula in euclidean spaceis striking.

* The residue of the present chapter is closely analogous to the treatmentof the corresponding euclidean problem given by Klein in his ' Einleitung in

GOOLIDOS Ij


The cones whose vertices are all at an arbitrary point, andwhich are circumscribed to a set of confocal quadrics, will

themselves be confocal, i.e. they will have foui- commontangent planes which touch the Absolute. Any two of these

cones will intersect orthogonally. This shows that the con-

gruence of lines tangent to two confocal quadrics will be

a normal one, the edges of regression of their developable

surfaces being geodesies of the cûadrics. These facts, well

known in the euclidean case, will be proved for the non-euclidean one in Chapter XVI. Notice that we get the

system of geodesies of a quadiic by means of its ao' commontangents with confocal quadrics. The difficulties which arise

for special positions, as umbilical points, need not concernus here.

The equation of the cone whose vertex is (F) and Whichcircumscribes the quadric (1) will be

0..S TT-j 0..3 y 2 p.O..S T7- irr-,^

i i • i *

Putting X = F+cZF we get the differential form

f f (c,-A)(c^.-X) -"•

Let us change this also to the elliptic form. We noticethat the coefficients of the expressions atA„ cZa. will be 0,

for the axial planes of the cones will be given by tan-gents to

A^ = 0, Aj = 0, A, = 0.

The co^ confocal cones form a one-parameter family all

touching the same tangent planes to the cone ds^ = 0. The

die hshere Oeometrie ', lithographed notes, Gsttingen, 1893, pp. 38-73, andStaude, ' Fadenconstruktion des Ellipsoids,' Mathemalische Annalm, vol. zx,1882. Staude returns to the subject in his Die Fokaleigenschajlen der Fldehmzweiter Ordnung, Leipzig, 1896. This book is intended as a supplement to theusual textbooks on analTtic geometry, and is somewhat prolix in its attemptsat simplicity.

xiii QUADRIC SURFACES 163

equation of one cone of the family may be thrown intothe form

\L,

where L^ is a function of X. Hence the genei-al form will be

' [ri(«i-v](i,-rt

It remains to find the value of L„—fi. It is clearly a poly-

nomial in powers of X, which vanishes only when X = X ,

for then only shall we have dX * = 0. We thus get

where A^ is a constant. Again, as two of these confocal

quadiics contain every line through the vertex, we musthave m = 1. Lastly, our expression is symmetrical in p, q, r,

hence a — A — A

We finally get for our cone

2 0^.. ' "^^ =0- (13)

For progress along an arc of a geodesic of X^ = const., wehave

^p-K0..b

{}^p-^)Uiî-^p)

"9-^'=0,

0..a

(^-^)n(î-v

so that the problem of finding the geodesies of a quadric

depends merely upon elliptic integrals. If we take X, = X,


we have double tangents to the surface, i.e. rectilinear

generators,

dk^ d\„

0..3

IliCi-K)

= 0.

The general differential of arc on a surface A, = const, is

d^_lA" "4

we have, then, for a distance along a generator

0..S

n («~VThis expression is independent of A,, whence

Thecn'e'ni 8. If from a set of

confocal central quadricsaone-

parameter set of linear genera-

tors be so chosen that all

intersect the same ao^ lines of

curvature of co^ confocal quad-rics of the system, then anytwo of these lines of curvaturewill cut congruent distances

on all of these linear

generators.

Theorem, 8'. If from a set of

homothetic central quadricsa one-parameter set of linear

generators be so chosen thatall touch co^ developablescircumscribed to pairs of

quadrics of the homotheticsystem^ then the tangentplanes to any two of lêse

developables will determinecongruent dihedral angleswhose edges are the givenlinear generators.

Theorem 8 may also be easily proved by showing that thegenerators of a set of confocal quadrics form an isotropic

congruence, whereof much more later.*

* The general theorem concerning isotropic congruences upon which thisdepends will be proved in Chapter XVI, where also will be foand a biblio-graphy of the subject.

XIII QUADRIC SORFACES 165

We now seek for the expression for the element of distanceupon a common tangent to two confocal quadrios \, A'.l

(Ap-Ag)dA^

^[n(«~v](^-v(^'-

= +

K)

(A_-A,)dA,(14)

K)

./ [ti (cf-v] (^-V (^'-

V

(Ag-A,)dA,= +

^^[ri (Ci-A,)](A-A,) (A'- A,)

di^p ^(^g-Ay)(A^-Ap

V ^' ^^

î> \ ^r

1 1 1

Va^-a,


ds dK(^-^)t =

0..S

n(«i-vy(\-A_)(\'-\„)

Multiplying through by (X-Xy), (A'-Ap), and summing for

p = 1. 2, 3

—UZS "^2

(\-K^)(\'-\^)0..S

n(ci-A3)

dAg. (15)

For a geodesic on A = Aj whose tangent touches A' we have

ds _ 1

(A--A3)(y-A3)^^^ ^,g^

ri(Ci-^3)

For a line of curvature common to A = Aj , A'= Ag

k~2 / ~o::5 «^3- (17)

It is now necessary to look more closely into the signs

of the radicals in (15). We know that, at least in a restricted

domain, three confocal quadrics will pass through each point.

In elliptic space one of these will be ruled, and the other two

not ruled ; assuming, of course, that we are dealing with the

case of central quadrics. In hyperbolic space, two possible

cases can arise in the actual domain. If the developable be

XIII QUADRIC SURFACES 167

real, two ruled, and one non-ruled hyperboloid will passthrough each point. If it be imaginary we shall have anellipsoid, a ruled, and a not-ruled hyperboloid.* Let ubconfine ourselves to this case, taking Xj as the parameter ofthe non-ruled hyperboloid, Ag as that of the ruled one, while A^gives the eUipsoid. The elliptic case will follow immediatelyif we suppress the word hyperboloid substituting ellipsoid.

In (15) let us assume that X refers to an ellipsoid, and A'

to a ruled hyperboloid. In two of the three actual axialplanes we shall have i-eal focal conies. There will be a realfocal ellipse which, looked upon as an envelope, constitutesthe transition between the ellipsoid and the ruled hyperboloid.It will be surrounded by all eUipsoids, and surround all ruledhjrperboloids. If we take a point in this axial plane, withoutthe focal ellipse, the ellipsoid and non-ruled hyperboloid will

subsist, the ruled hyperboloid, looked upon as a point locus,

will shrink into the plane counted doubly. The other real

focal conic will be a hyperbola, and will serve as a transition

between the two sorts of hyperboloids, looked upon asenvelopes. It will surround the non-ruled hyperboloids, butbe sun-ounded by the niled ones. The plane counted doubly,

will replace a non-ruled hyperboloid for each point withoutthe hyperbola. If a point be taken in the remaining axial

plane, this plane, counted doubly, will replace a non-ruledhyperboloid for each of its points. Similar considerations

will hold in the elliptic case.

Once more, let us look at the signs of the terms in (15).

dki will change sign as a point passes through an axial plane

that counts doubly in the A{ family, or when passing alonga tangent to one of these surfaces, the point of contact is

traversed. On the other hand we see &om (14) that whenÂj changes sign, the radical associated with it in (15) changes

sign also, and vice versa. The radical associated with dK^

will change sign as we pass through a point of the axial plane

with an imaginary focal conic (which we shall call v,), andfor a point of the axial plane ttj of the focal hyperbola, whichis without this hyperbola. The radical with dK^ îU change

sign for points of tt, the plane of the focal ellipse without this

curve, or points of Wj within the focal hyperbola. The radical

with dA, will change sign for points of irj within the focal

ellipse.

We next suppose that a loop of inextensible thread is slung

about an ellipsoid A, and a confocal, ruled, one-sheeted hyper-

boloid A', and pulled taut at a point P. The loop is supposed

* See the Author's ' Quadric Surfaces', p. 165.


to surround the ellipsoid, so that it winds partly on each of

the portions of the hyperboloid, which, in a restricted domain,

are separated by the ellipsoid. The form for the element of

length throughout the whole string wiU be that given by (15).

For when we pass from the ellipsoid to the hyperboloid wepass along a geodesic whose tangent touches both surfaces,

and this will be true throughout the continuation of that

geodesic, for a geodesic is traced by a line rolling on a quadric,

and touching a confocal one. The same form of distance

element will hold for the rectilinear parts of the loop. Wesee, moreover, that two, and only two surfaces, of a confocal

system will touch any line ; hence X and \' are the only twowhich will touch the rectilinear parts of the loop. Lastly,

let us limit oui-selves to those regions of the plane where thevarious portions of the loop may be named in order : straight,

hyperboloidal, ellipsoidal, hyperboloidal, ellipsoidal, straight.

The constant length of the thread may be written

F^dX^ + F^dX^ + F^dX^.

We see that F^ can never vanish, for \ and V are the pai-a-

meters of an ellipsoid and ruled hyperboloid respectively,while Aj refers to a non-ruled hyperboloid. It will becomeinfinite four times, twice when the loop passes ^2 the planeof the focal hyperbola, and twice when it passes Wg. We may,however, integrate right up to these limits, and, as we haveseen, d\^ changes sign with the radical. We thus have

f'^,r', r'2 [•', fc, pA,

F^d\s=\ F^d\^-\ F^d\s+\ F^dKs-\ F^dK^+l F^dX,-K JA, Jc, Jtj Jo, Jrj

= 4 VgdAg = const.Jf,

We may approach the second integral in the same spirit.

F^ will become infinite twice when the loop passes the planeof the focal ellipse w,. It will vanish throughout those twoportions of the loop that lie on the ruled hyperboloid X^ = X',

and these two are separated by an intersection with Vi. Wehave then

f 'F^dX^ = [ F^dX^- f 'F^dK, + r F^dX^- r>,d\, + fVdAj-'A, jAj Ja' Jc^ Jx' Jc,

-JA'

F„dX„ = const.

xin QUADRIC SURFACES 169

We must, in conclusion, consider the first integral. It willnever become infinite, but will vanish along those two portionsof the loop which lie on the ellipsoid X = Xj. We havetherefore

:

Fid\^=\ J\d\i- Jf,dA, = 2 J!\dXi = 0(Xi).Ja, Ja, Ja Jai

We have therefore, since the first two integrals and thesum are constant,

(Xj) = const,

and the locus of the moving point is an ellipsoid. Lastly, let

the ellipsoid and hyperboloid shrink down to the focal ellipse

and focal hyperbola respectively, we have in the limiting case

:

Theorem 9. If an eDipse and hyperbola in mutually perpen-

dicular planes pass each through two foci of the other, andif a loop of inextensible thread be slung around the ellipse

and pulled taut at a point P in such a way that it meets

the two curves alternately, then the locus of P will be anellipsoid confocal with the given ellipse and hyperbola.

CHAPTER XIV

AREAS AND VOLUMES

The subjects area and volume oflFer some of the most

striking points of disparity between euclidean and non-

euclidean geometry.* A first notable diflFerence ariaes from

the fact that, in the non-euclidean cases, two different func-

tions of a triangle appear to play the r6le of the euclidean

area. The first is present in the analoga of those formulae

which give the area in terms of the sides and angles ; the

second appears when the area is defined as the limit of a

sum, i.e. as a definite integral. We shall reserve the namearea for the second of these, giving to the first the nameamplitvde.^

Let us, as in elementaiy geometry, use the letters A, B, Cto indicate, either the vertices of a triangle, or the measures

of its angles. We assume that these points are real, and,

in the hyperbolic case, situated in the actual domain. Weshall define triangle as in Chapter II. We might carry

through the same sort of work for any three points, but,

as we saw in the closing pages of Chapter VII, we should

thereby be compelled, in the hyperbolic case at least, to

introduce certain very delicate considerations as to algebraic

sign, not only in our analytic expressions, but even in the

trigonometric formulae first introduced in Chapter IV.

We begin by rewriting IV. 9

. b . c , be a— sin r S11I7; cos A = cos T cosr — cos r-

This foimula, established for one region, is seen at once to

hold for all the others.

* For a bibliographical account of the subject-matter of the present chaptersee the dissertation of Dannmeyer, Die OberJlSchen- und Yvlumenbtrechnwng furLobatsche/skijsche Sdume, Giittingen, 1904.

f The concept amplitude of a triangle, and the various trigonometric iden-tities connected with it, are taken directly from an admirable paper byD'Ovidio, ' Su varie questioni di metrica proiettiva,' Atti della R. AceademiadeUe Scieme di Torino, vol. xxviii, 1893. Unfortunately the author gives, p. 20,an incorrect formula for the volume of a tetrahedron.

CH. XIV AREAS AND VOLUMES 171

. b . c . .

smy- sin 7 sin ilk k

Bin' T sin^j^— cos^ r cos* t + 2 cob t cos t cos j —cos'' t

-cob'j-A;

-cos^T — cos^'r + 2 cos r cos f cos

The right-hand side is symmetrical in the three letters a, h, c,

so that we may write

sinh . c . . . c . a . -n . a . . ^T Sin T sinA = Bin-T sin ^ sin£ = sm t sin r sm (7

k'

, e b1 cos T cos T

cCOSj^

cos T COS

acos

J (1)

k k

In the real domain, if the measures of sides and angles betaken positively, the left side is essentially negative in thehyperbolic case, and positive in the elliptic, so that theradical on the right must be chosen accordingly. It will

vanish only when the three points are collinear (under therestrictions made at the outset of this chapter), and shall becalled the Sine Amplitude of the triangle, written sin (ABC).

Let the reader show that if the coordinates of A, B, G be

{x), (y), (z) respectively

(axe) {any) (xz)

{yx) (yy) (yz)

{zx) (zy) (zz) xyzI

-/{xx) -/{yy) V{zz) '^{xx) '/{yy) '/{zz)

We may rewrite (1) in the form

sin A sin B sin (7 _ sin (ABC)

.a . b . csin T sm T sm

k

. a . b . csin r sin t sm :rk k k

(2)

(3)

'4 ""'& " k

If A', B', C be the points where the sides of the triangle

meet the perpendiculars from the vertices, we have

. a . AA' . b . BF . c . GO'I r sm —r- = sin r sm -j— = sin j sin -^j-

k k k K k ksin r sm —r- = sin y, sm —j^ — sin j_ sin -^ = sin {ABC). (4)

172 AREAS AND VOLUMES CH.

We see at once the close analogy of the sine amplitude of a

non-euclidean triangle to double area of a euclidean triangle.

Let the reader show that

Lim. p = 0, k^ sin [ABC) = 2 Area A ABC.

A function cori'elative to the sine amplitude may beobtained from the correlative formula

sin B sin C cos =- — cos B cos (7+ cos A.

sin £ sin C sinr = sin C sin J sin r = sin J. sin B sin r

1 cos (7 cos 5 icos C 1 cos A 1

cos B cos A 1I

= sin (abc). (5)

This > in the elliptic case, pure imaginary in the hyper-bolic

.a . b . csin 7- sm-r sinj- . , , >

« k _ k _ sin {abc)

sin A ~ sinB ~ sinG~ saiA sinB sin C (6)

. . . ^^' . ^ . BB' . ^ . CC .,,,,„,sm il sin —r— = sin £ sin -:— = sm U sin —j— = sm (abc). (7)k

.a . b . csm7 sin 7- siny . , . t,„.

k _ k _ k _ sm (ABC)sinA ~ sin B ~ sin C ""

sin (a6c)(8)

. , , . sm^ (ABC) . , . „„. sin^ (ahc)sm (aic) = ^

. ' , sa:i{ABC)=-— . / '. ^ . (9)^ ' " '" " ' sm .4 sin 5 sin (7 ^

'a . b . csinrsmTSinr

If a + 6 + c = 2s,

cos j1 =

sinf^ =

a becos 7 — cos T cos rk k k

. b . c'

. 8—b . 8—Csm —;— sm -=

—

k k

sm r sm r

XIV AREAS AND VOLUMES 173

cos Â —

ctn Â =

. s . 8—

a

. b . csin T Bin T-

. 8 . 8—

a

sm t; sin —r—

Bin8-b . 8— C

sin (ABC) = 2 /si8 . 8—a . 8— 6 . 8— C

BUiTsm- , sin—p-sm-jTV- (10)

In like manner, let us put

A + B + C = 2(T.

Bin J O _ r— cos <r COB (o- —4)"]^

'i~L sin £ sin C J '

a _ rcos (<r—B) cos (<r— (7)

sin B sin (7]'

'^4"~L —co8o-cos(<r— a) Jctn

sin {abc) = 2 -/—cos o- cos (a-—A) cos (o-— 5)cos (»— C). (11)

sin^^sin|5sin^(7 =(sin-

8—

a

Bin-8—6 . 8-

-")

. a . 6 . csinrSinjrSinTr

sin-r = sin (abc)

— cos (7 =

k 4sin^^sin^£sin^(7

sin (ABC)

4oos|jcos|jcos^^

(12)

(13)

It should be noticed that the denominator on the right of

equation (13) is essentially positive. The numerator is

negative in the hyperbolic case, as we have already seen,

174 AREAS AND VOLUMES ch.

but here also <r < ^ and cos o- > 0. In the elliptic case the

numerator is positive but <» > o' ^^^ o" < 0.

In Chapter III we defined as the discrepancy of a triangle,

the absolute value of the difference between the sum of the

measures of the angles and -n. Let us now define as the

excess of our triangle the expression

e = A+B + G-v.

This will have the same sign as r^ > the measure of curvature

of space. We have

• « sm {ABO) ,.^.sm- = — cos o- = i ji (14)2

. , a ,b4 COB J r- cos ^ -r cos ^ T

case where the trianj

_. sin(4jBC) .,. / ,o ,^ ,c\JLiim. = 4 lim. (cos ^ - cos | -r cos f rj

Passing to the limiting case where the triangle becomesinfinitesimal, we have

sm^

= 4

Um. e = ^lim.(ilJSC)

= ;rn lim. he sinA

= -î^m.aAA'.

Theorem 1. In an infinitesimal triangle the limit of theratio of the excess to the product of the eudidean area andthe measure of curvature oi^space is unity.

Let us next examine the infinitesimal quadrilateral, whosevertices axeA,B,G,D. AB and CD shall intersect in H (actualor ideal) while AC and BD intersect ia K\ the latter twopoints remaining at a finite distance fi-om A, B,G, D.

. ABBin -T— . „

k amK. BK sinil'

k


. AB . AC . .

• tn An\ Sin -7— sm -;— sin A,. sax (GAB) ,. k klun.

. )r,Aii\ — I'ln- = =%m{DAB)

. DB . DG . ,,sm -T-sin -?— sm x>

= lim.

sm -r-sm -T-sin D

AB.AG.aiaABB. DC. Bin

D

= 1.

We shall define as the area of an infinitesimal triangle

the common value of P times its excess, its half-amplitude,

and the eudidean expression for its area.

Theorem 2. If the opposite sides of an infinitesimal

quadrilateral do not intersect in points infinitesimally nearthe vertices, the limit of the ratio of the areas of the triangles

into which it is divided by a diagonal is unity.

The sum of these two infinitesimal areas shall be called

the area of the infinitesimal quadiilateral ; it will be equal

(always neglecting infinitesimals of higher order) to the

product of two adjacent sides multiplied into the sine of the

included angle.

Suppose now that we have a region of the plane^ connexright up to the boundary, which is limited by one or moreclosed curves, and let this be covered by a network of in-

finitesimal quadrilaterals of the sort just described. Let the

area of each of these be multiplied by the value for a point

therein of a continuous function of the coordinates of the

point. The limit of this sum as the individual areas tenduniformly toward zero shall be called the surface integral of

the given function for the given area. The proof of the

existence of such a limit, and its independence of networkemployed will be identical with that used in the correspond-

ing euclidean case, and need not detain us here.'"

Definition. When the surface integral of the function 1

exists over a region of the plane, that integral shall be defined

as the area of the region.

Theorem 3. The area of a region of a plane is the sum of

the areas of any two regions into which it may be divided

provided that these two have no common area.

This follows immediately from the definition given above.

As an application of these principles let us determine the

» Conf. e.g. Ficard, Traits iCAnatyse, first ed., Pavis, 1891, vol. i, pp. 83-102.


area of a triangle. It is the limit of the sum of the areas of

a network of infinitesimal triangles, or by (1) the limit

of the sum of P times their excesses. Now it is perfectly

clear that if a triangle be divided in two by a segment whose

extremities are a vertex and a point of the opposite side, the

excess of the original triangle is the sum of the excesses of

the parts, and we may establish our network by a repetition

of this process or division, hence *

Theorevi 4. The area of a triangle is the quotient of the

excess divided by the measure of curvature of space.

Let us give a second demonstration of this fundamental

theorem with the aid of integration. It will be sufficient to

do so in the case of a right triangle, and we shall take a right

triangle with one angle at C the intersection of x^ = 0, aig = 0>

the right angle being at £ a point of the axis ajg = 0. Wemay introduce polar coordinates

—' = fc tan r cos <t>, — = k tan rsin i

Xo & a;, k

the elements of arc along . (15)

rnJ. / R\

kl sin T dr = k'il— COS -r)

tanI = tan^ sec <(,. (Ch. IV. (6).)

It coa<b

/cos^c^ + tan''^k

Remembering that the limits for are and C

* It is surprising to see how unsatisfactory are the proofe usually given for

this, the best-known theorem of non-euclidean geometry. In I^rischauf,

BlanetUe der absoluten Oeometrie, Leipzig, 1876, will be found a geometrical proofapplicable to the hyperbolic case but not, so far as I can see, to the elliptic,

and the same remark will apply to the book of Iiiebmann, cit. Manning,loc. cit., makes an attempt at a general proof, but the use of intuition is

scarcely disguised. In Clebsch-Lendemann, Vorksungen Hber Oeometrie, Leipzig,

1891, vol. ii, p. 49, is a proof by integration, but the analysis is unnecessarilycomplicated owing to the fact tha^ apparently, the author overlooked theconsideration that it is sufficient to prove the theorem for a right triangle.


Area = J5?rd^-P /'_-=îiL=^ .

•'0 Jo I 57^

The first integral is ^2(7^ If, further, we put sin <^=a;,

J I W = «^'L« COS ^J+ const.

Hence our second integral will be

-i«|sin-»rsin<^cosîr.

This vanishes at the lower limit. On the other hand byChapter IV. (7)

cos j4 = sin (7 cos _ ,

h '

our second integral becomes

Area = kîA+B+ C-v). (16)

Two regions with the same area may, naturally, have verydifferent shapes. There are, however, three simple cases

where the equivalence of area is immediately evident. First,

where the two figures are congruent ; second, when they are

composed of the same number of non-overlapping sub-regions

(i. e. sub-regions no two of which have in common a region

which has an area) congruent in pairs ; third, where by the

adjunction of pairs of mutually congruent non-overlapping

sub-regions to them, they may be transformed into congruent

regions. In this latter case they may be said to be equivalent

by completion.*

Definition. Given n successive coplanar segments (A^Â.,),

(AjfAj^^^), (i4»-i-4i) so situated that no line other than one

through a point Ai can contain points of more than two of the

segments ; the assemblage of all points of all segments whose

* The term equivalent by campleHon ia borrowed from Halsted, loc. cit., p. 109.

The distinction between equivaUmt and equivalerU by completion is, I believe, due

to Hilbert, loc. cit., p. 40. For an admirable discussion of the question of

area see Amaldi, in the fifth article in Enriques, (iuestioni rigwurdanti la geome-

tria Oementare, Bologna, 1900.

COOLIDSB M


extremities are points of the given segments shall be called aconvex pUygon or, more simply, a polygon. The definition

of sides, vertices, and angles is immediate. If one vertex, say

Ai, be connected with all the others, the polygon -will be

divided into n—Z triangles, no two of which nave in commonany area. The area of the polygon will thus be the sum of

the areas of these triangles. We may convince ourselves of

the compatibility of these statements as follows. A triangle

is certainly a polygon, and if a polygon of to— 1 sides exist,

we may easily enlarge it to have n sides by taking an addi-

tional vertex near one side. On the other hand, if a polygon

of n— 1 sides may be divided up in the manner suggested,

it is immediately evident that one of n sides may be so

divided also.

Theorem 5. The area of a convex polygon is the quotient of

the excess of the sum of its angles over {n— 2)v divided bythe measure of curvature of Space.

Let the reader show that the area of a proper circle is

2ir/fc2(l-C08^)- (17)

The total areas of the elliptic and the spherical planes will berespectively

In the hyperbohc plane regions may be found having anydesired area.

Our next undertaking shall be to see how far the methodswhich we have established for studying areas are applicable

in three dimensions. We shall begin, as before, wiui ampli-tudes, following, however, an analytical rather than a ti-igono-

metric method.Let the vertices of a tetrahedron, as defined in Chapter II,

be A, B, C, D with the coordinates (a;), (y), {z), {t) respectively.

The opposite faces shall be a, /3, y, 8 with coordinates (u), (u),

{w), (o)), so that, e. g.

r(a)X) = (Xaryz).

We shall define as sine amplitude of the tetrahedron

e,m.(ABCD) = AA BB 00 DDcos —J— cos -T— cos -T- cos ^—

k k k k

\{^){yy)(zz){tt)\''^

-/{xx) -/{yy) ^z) ^/{tt)


^ \^^\ _ ns)Vixx) Viyy) V{zz) V{tt)

'

We shall give to the radicals involved such signs thatk sine amplitude shall have the sign of ¥. Recalling theconcept of the moment of two lines introduced in Chapter IX,we get

An fij)

sin-J-

sin -^ (Moment AB, CD) = sin (ABGD). (19)

sin(il£C) = -lMM^^.•/(asc) •/(2/y) i/{zz)

Let A', B', C, ly be the points where perpendiculars fromthe vertices of a tetrahedron meet the opposite faces. Then

sin^= I'^^^ 1

^ VpI{xxY^) {zz)

1

-^

'

_sin {BCD) sin —r- = sin (CDA) sin -r- = sin (DBA) sin -r-

= sin (ABC) sin^ = sin (ABCD). (20)

If we mean by âj3 the dihedral angle of these two faces

(uv)

(ty) (tz) (tt)

-/(uu) V{w)

(oey) (ass) (xt)

(zy) {zz) {zt)

/ Z\(xx){yy){zz){tt)\ I

V 2) {XX.) V

sin(ilBCI>)sin^

sinâ/j = -;-

3|(a3a!)(2/3/)(zs)(«)|

^(2/2/)

sin(-B(7X>)sin(il(7Z))'

sin {BCD) sin (il(7i)) ?iB-^ = sin (ilSCD). (21). AB

The geometry of lines through a point is an example of the

M 2


geometry of the elliptic plane, •where A* = 1. We may thus

speak of the sine amplitude of a trihedral angle

1

sin (AB, AC, AD) =

(ft) (tx)

(xt) (xx)


The analogy between the sine amplitude and the sextupleof the eudidean expression for the volume appears even moredistinctly in the infinitesimal domain.

Lim. sin {ABC) = ÂB .AG.am4-BAG

= jÂrea A ABC.

Lim. sin (ABCD) = lim. (ABCD)

= p Vol. tetrahedron ABCD. (28)

Following our previous analogy, suppose that we have six

planes, no three coaxal, passing by fours through four actual

or ideal, but not collinear points. Let the remaining inter-

sections be at a finite distance from the three chosen points,

but infinitesimally near one another. An infinitesimal

region will thus be formed, on the analogy of a euclidean

parallelopiped, which may be divided into six tetrahedra of

such sort that the limit of the ratio of the sine amplitudes, or

of the euclidean volumes, of any two is unity. Six times the

euclidean volume of any one of these tetrahedra may be defined

as the euclidean volume of the region.

So far the analogy between two and thi-ee dimensions hasbeen sufficiently good. Each time we have had a function

called sine amplittide con-esponding in many particulars to

a simple multiple of the euclidean area or volume, and ap-

proaching a multiple of the area or volume as a limit,

when the figure becomes infinitesimal. Li the plane there

appeared, besides half the sine amplitude and the euclidean

area, a third expression, namely, the discrepancy or excess.

In three dimensions this function is, sad to relate, entirely

lacking; that is to say, there is no simple function of the

measures of a tetrahedron which possesses the property that

when one tetrahedron is the logical sum of two others, the

function of the sum is the sum of the functions. It is the

lack of this function that renders the problem of non-

eudidean volumes difficult."'

Suppose, in general, that we have a three dimensional

region connex up to the boundary, and that we divide it

* It is highly interesting that in four dimensions a function playing the

rdle of the discrepancy appears once more- See Dehn, ' Die eulersche Formelin Zusammenhang mit dem Inhalt in der nicht-euklidischen Geometric,'

Xathematische AntuUen, vol. Izi, 1906.


into a number of extremely tiny tetrahedra. The limit of

the sum of the euclidean volume of each, multiplied by the

value for a point therein of a continuous function of the

coordinates of that point, as all the volumes approach zero

uniformly, shall be called the volume integral for that region

of that function. The proofs for the existence of that volume

integral, and its independence of the method of subdivision,

are analogous to those already referred to for the surface

integral. In particular, the volume integral of the function

unity shall be called the volume of the region. Two regions

will have the same volume if they be congruent, made up

of the same number of parts, mutually congruent in pairs,

or if by the adjunction of such pairs they may be completed

to be congruent.

If the umiting surface of a region be made up of a series of

plane surfaces, and if no line, not lying in a plane of the

surface, can contain more than two points of the surface, then

it is easy to show that the region may be divided up into anumber of tetrahedra, and the problem of finding the volume of

any such region reduces to the problem of finding the volumeof a tetrahedron. This problem may, in turn, be reduced

to that of finding the volume of a tetraJiedron of particularly

simple structure. To begin with, we may assume that there

is one face which makes with the three others dihedral angles

whose measures are less than -> for the bisectors of thetil

dihedral angles of the original tetrahedron will always divide

it into smaller tetrahedra possessing this property. The per-

pendicular on the plane of this face, from the opposite vertex,

will, then, pass through a point within the face, and, with the

help of this perpendicular, we may subdivide into three

smaller tetrahedra, for each of which the line of one edge is

pei-pendicular to the plane of one face.

Consider, next, a tetrahedron where the line of one edgeis indeed perpendicular to the plane of a face. There are

two possibilities. First, in the plane of this face neither of

the face angles whose vertex is not at the foot of theperpendicular is obtuse; secondly, one of these angles is

obtuse. (The case where both were obtuse could not occurin a small region.) In the first case we might draw a line

from the foot of the perpendicular to a point of the opposite

edge in this particular iace, perpendicular to the line of

that edge, and thus, by a familiar theorem in elementarygeometry, which holds equally in the non-euclidean case,

divide the tetrahedron into two others, each of which possesses


the property that the lines of two opposite edges are per-pendicular to two of the faces. These we shall for the momentcall si/mplest type. In the second case, from the vertex of theobtuse angle mentioned, draw a line pei-pendicular to the line

of the opposite edge in this particular face (and passingthrough a point within this edge), and connect the intersectionwith the vertex opposite this face. The tetrahedron will bedivided up into a tetrahedron of the simplest type, and oneof the sort considered in case 1. We have, then, merely toconsider the volume of a tetrahedron of the simplest type.

Let the vertices of the tetrahedron h& A,B, G, D, where ABis perpendicular to BCD and BC perpendicular to ABC. Let aplane perpendicular to AB contain a point .Sj of {AB) whosedistance from A shall have the measure x ; while this planemeets {AG) and {AB) in Cj and Dj respectively. The volumeof the region bounded by this plane, and an adjacent one of

the same type and the three faces through A, will be dx, mul-tiplied by the surface integral over the A B^ (7, D^ of the cosine

of the A"" part of the distance of a point from B^. (Cf. Ch.IV. (2).) This integral takes a striking form.*

Let the distance from B-^^ to a vaiiable point P of the

triangle be r, while <^ is the measure of 4~ î B^P. We wishto find

k\ \ sm. T COB rdrd^.

Let B^P meet {C^D^ in E^. The limits of integration for r

are and B-Ê^ ; hence we have merely to find

i^^r^W.. ^B^

2J0sm^-^^'^-

Now CiZJj is perpendicular to B-^C.^, hence

tan (^ sin^ = tan^,* The integration which follows is a very special case of a much more

general one for » dimensions given by SchlSfli, Thearie der vieHfachm Kontinuitat,

Z&iieh, 1901, p. 646. This paper of Schl&fli's is posthumous ; it was originally

written in 1855, when the science of non-eucUdean geometry had not reached

itB present recognition. It is very general, extremely difficult reading, andhampered by a fearful and wonderful terminology, e. g. our tetrahedron of the

simplest type is a special case of an Ariiothoscheme. It is, however, a striking

Itiece of geometrical work. Schl&fli gives a shorter account of his work in

his ' Reduction d'lme int^grale multiple qui comprend Tare d'un cercle et

I'aire d'un triangle sph^rique comme cas particuliers ', LionviUe's Journal, vol.

zxii, 1866.


cos^-l = COS ^7-* COS -^,_ ^ tan— = tan—j^-' sec ^.

(Oh. IV. (5), (6).)

sin'B,E,,^ 1 . -B,Ci

k ^~kOur required integral is then

dip = T Bin -ij-^ d E^G^.

IIsmîC\^^_fc. MldE,G, = âm^ • C,Ak

Let the reader note the astonishing feature of this result,

namely, that it involves one side of a triangle directly, andanother trigonometrically.

Let the measure of the dihedral angle whose edge is (Ci-Di)

be d, this will also be the measure of ÂG^B^ which ia the

plane angle of the dihedral one.

ÂB^cos = cos—jT—* sm^BAG,

1 ABsinOdO = Y sin —r^ sin 4-BAGdx

1 . AB[ «"^-V= k'^-k--—^'^sin—rJ

k

= r sin tf sin —V- dx.

We thus get for our volume the strange formula *

Vol = ~\G;^^de. (29)

We can easily express this integral in terms of 0,

^^' = sm-^ ta.n 4. BAG = a Bin^,

AGcos -T^ = ctn ^£^Cctn tf = 6 ctn 0.

VoL = ^ I

tan-i [a /!- 6^ ctn« 0]dO. (30)

* See Schlafli, Riduetion, p. 381 , where it is stated that this integral cannotbe evaluated by integration by parts. This same integral was discovered,

apparently independently, by Bichmond, ' The Volume of a Tetrahedron in

Elliptic Space,' Quarterly Journal of Mathematics, vol. xxziv, 1902, p. 176.


This formula apparently represents about as close anapproach as can be made towards finding the volv/me of this

tetrahedron, for, in the general case,* it does not seem possible

to effect the quadrature in terms of elementary functions.

If a right triangle be rotated completely about one of thesides adjacent to the right angle, the figure so generated shall

be called a cone of resolution. The volume within the surface

may be found as follows. Let the vertex of the cone be Aand the centre of the base 0, while P is a point within the

cone. Let Q be the intersection of {AO) with a perpendicular

from P, while the base circle meets the plane AOP in B.

{AB) shall meet PQ in R. Let us also write

AB = s, AR=r, AO = h, 4-OAB = 0.

J'QHrh ri-r Qp Qpsm^Qos^dAQdQPd^Jo Jo "^ '^

= 2;rijT^sin^cosH^dlQ . dQP

tan:^ = tan ^ cos 6. (Ch. V. (6).)

dAQ =

Tcos sec^ r dr

k

l + coB^fltanVK

. QB ^ asm -'rr- = sm rsm Q.

Vol. = wfc2sin2flcosfl

Ttan* T-

k

l+cos^fltan*^

-dr.

rPut tanT = a;.

* Schlafli, YieLfaOix Kmtinuiiat, p. 95, gives a formula for the special case

where the sum of the squares of the cosines of the dihedral angles is equal

to unity. The proof is highly intricate, and not suitable to reproduce here.

186 AREAS AND VOLUMES ch. xiv

h

Vol. = ^fc« cos e sin- eJ^ (i + a;.)(n.a,^cos^fl)

A

= wJk=* cos 6\

-tan"^(a;cos e)-taD.-'^xLcosS Jo

= •^^^[^-8 008 0].* (31)

To find the volume within a proper sphere, where the

distance from the centre to every point of the surface has the

constant value R, ,

VoL = F siD?^smedrded

PR ».

Jo k

Let the reader show that the total volumes of elliptic andof spherical space, where k = 1 will be, respectively,

* This formula is given without sufficiently detailed proof by Frischauf,

loc. cit., p. 99. A tedious demonstration was subsequently worked out byTon Frank, ' Der KOrperinhalt des aenkrechten Cylinders und Kegels in derabsoluten Geometrie,' Ortttierts Anhiven, vol. lix, 1876.

CHAPTER XV

INTRODUCTION TO DIFFERENTIAL GEOMETRY

The task which we shall undertake in the present chapteris to develop the differential geometry of curves and surfacesin non-euclidean space.* We shall introduce a notahle sim-plification in our work by abandoning homogeneous coor-

dinates, and assuming that

{xx) = ¥. (1)

In the elliptic case we shall take a;,, ^ ; in the hyperbolic,

;»(, = J Xj ^ for all real points.

Of course in exceptional cases, where we wish to include

points of the Absolute or beyond, this proceeding is notlegitimate ; we shall therefore assume, unless we specifically

state the contrary, that we. are limiting ourselves to a real

region, where no absolute or ultra infinite points are included

in the hyperbolic case. We shall, furUier, have for the

distance of two points {x), (x').

(2)

d (xx') . „d"°^]k

= V' ^"^1 = ¥When a;/= x^ + dx^ we have for the square of the differential

of distance

l^^ = rfs2= {xx){dxdx)-(xdxf

A* A*

{x-{-dx, x+dx) = k^, (xdx) = —^{dxdx),

d^ = (dxdx). (3)

We shall mean by an analytic curve, such a curve that the

coordinates of its points are analytic functions of a single

variable. The formulae developed in this chaptei- will hold

* The developments of this chapter follow the general scheme worked out

for the euclidean case in Bianchi-Lukat, Vmrlesungen uber DiffereTitialgeometrie,

Leipzig, 1899, Chapters I, III, IV, and VI. In Chapters XXI and XXII of the

same work will be found a different development of the non-euclidean case.

It is, however, so general, yet so concise, as to be scarcely suitable to serve

as an introduction to the subject.

188 INTRODUCTION TO CH.

equally well under the supposition that the functions and

their first three partial derivatives exist and are finite in our

region, but the gain in generality is of little interest to the

geometer, and we shall assume from here on that when wespeak of curve we mean analytic curve.

Let us imagine that at a chosen point of a curve, say P, a

tangent is drawn. We shall take two near points P' and P"on the curve and tangent respectively, so situated near P and

on the same side of the normal plane that PP'= PP". Thenwe shall define*

,. 2P'P"iim. -==r >

as the curvature of the given curve at that point. If wecompare with Chapter XI. (2), and define as the osculating

circle to a curve at a point, tiie limit of the circle throughthat and two adjacent points, we shall have

Theorem, 1. The curvature of a curve at any point is equal

to that of its osculating circle, and is equal to the absolute

value of the product of the square root of the curvature of

space and the cotangent of the «"> part of the distance of each

point of the circle from its centre.

Let us now suppose that the equations of our curve are

written in the form

Then for a point on the tangent we shall have coordinates

X, = \{xi{Q+{t-t,)xaQlTo get the value of \

{XX) = xx = k'^, {xx') = 0.

Developing by the binomial theorem, and rejecting powersof (t—tg) above the second

* This definition is taken from Bianchi, loc. cit., p. 603. It is thereascribed to Voss.

XV DIBTERENTIAL GEOMETRY 189

Subtracting from the series development of x^, we get for

our curvature - •

P

1[K'a:") + p (a;^") (^'a=') + p {x'x'r {xx)j

Theorem 2. The square of the curvature of a curve is thesquare of its curvature treated as a curve in a four-dimensionaleuclidean space, minus the measure of curvature of the non-euclidean space.

It wiU be convenient to consider, besides our point («),

three other points allied to it. (t) shall be orthogonal to [x)

and on the tangent, (z) orthogonal to {x) on the principal

normal, and (^) orthogonal to {x) on the binormal. Thesethree will replace the dii'ection cosines of tangent, principal

normal, and binormal, which figure so prominently in the

euclidean theory. In hyperbolic space these points lie withoutthe actual domain to which we suppose {x) confined.

{xt) = ixz) = (x^) = (tz) = {to = (zO = 0.

If a point trace an infinitesimal arc ds, the angle of the

corresponding absolute polar planes ^^ a /p" *

We shall, hereafter, take as our parameter on the givencurve 8, the length of arc, so that

As {t) lies on the tangent, its coordinates will be of the form

ti = lxi+mxi,

{tt) = (xx) = k% {tx) = 0, (aKB') = 0,

ti = kx/. (5)

For the point (z) we shall have

Zi = XXi+ llXi+VXi",

(zx) = (za^) = (xa^) = {a/af)+ {xx") = 0,

(zz) = {xx)=k\ (x'x') = 1.


^' ~ ^¥{x"x")-l

'

Zi = ^(Xi + k^Xi")- (6)

To determine ^ we shall have the conditions

^, = p±_\yxx'x"\. (7)

We shall define the torsion of our curve as the limit of the

ratio of the angle of two successive osculating planes to the

differential of arc. We thus get

T~k da '^^'

Reverting to our formulae (5) and (6)

dtf Zi Xf .rt.

da- p~ k^^'

{xo = (x'i) = {:^'i) = (xe) = {x'o =m

=

0.

Hence dPi ,' = IZj

ds"»>

or, more specifically ^f- z-

We have also , . , n / , \ i i^ n(ass) =z (xz) = (xz) = (zz) = 0,

^- k ii nnd8~~ p~ T ^^^^

The reader will see at once that (9), (10), (11) are the

analoga of Frenet's formulae for eudidean curves.

We have, so far, overlooked the question of the sign of thetorsion, but that is well determined firom the above foimulae,and it is important now to find the geometric difference

between the case where the torsion is negative, and thatwhere it is positive. We shall carry through the work for

the elliptic case only, the hyperbolic may be treated in the

same way, but it is wiser there to replace the coordinates

(a;) by (x).

As before we shall choose s as the independent variable,

so that

{xx) = k', {xaf) = {x'a^') = 0, {x'x') = -{xa^')=l.

XV DIFFERENTIAL GEOMETRY 191

The sign of «,• (which may be ideal) will be found from (5),that of Zi from (6), and that of f,- from (7), while the signof T will be given by (10).

The equation of the plane of the tangent and binormal

Putting in the coordinates of a near-by point of the curve.

¥- (As)'"i + k\x"af')^ = F {Asf

2 '2 "" " ' 2 p-

and this is essentially positive, so that, in general, the curvewill not cross this plane here. Again, we see by (6) thatwe may give to a point on the principal normal close to {x)

the coordinates /»-„'/

Substituting in the equation of the plane we get

so that this will lie on the same side as the curve if e > 0.

Let us call positive that part of the curve near our point for

which As > 0. The positive part of the tangent shall be thatwhich lies on the same side of the normal plane as thepositive part of the curve,- while that part of the principal

normal shall be called positive which lies on the same side of

the plane of tangent and binormal as does the ciuve. Let usfind the Plueckerian coordinates of a ray from x^+x/As onthe positive part of the tangent to x^+eXi" on the positive

part of the principal normal. We get

P« = «Xi Xj

+ AsXiXj + eAs!'^'-„\

Xi xfIn like manner for a ray from («) to a point on the positive

part of the curve

Xi+ a;^ AjS + Xi —2— + a:,- -gj-

.

we get

<lu = î«{A^Bf

, (V)!3!

The relative moment of these two rays, as defined at the

close of Chapter IX, will be

192 INTRODUCTION TO ch.

The factors outside of the determinant are all, by hypothesis,

positive, so that the sign depends merely upon that of the

p(i<JC )

determinant, and this by (7) is equal to , •

^°^(ix")ô, (ix"') = -(ex").

Hence the relative moment will have the sign of

Theorem 3. The toraion at a general point of a curve is

positive when the relative moment of a ray thence to a point

on the positive pai-t of the curve, and a lay from a point onthe positive part of the tangent to one on the positive part of

the principal normal is negative ; when the latter product is

positive, tne torsion is negative.

Intuitively stated this means that the torsion is positive

when the curve resembles a left-hand screw, otherwisenegative.

We shall next take up the evolutes of a curve. Let (x)

be a point of an evolute. Then

« _ . Sr-Xi = COS T î — sin T ti,

dxi . 5 Zi-^ = — sm r - '

CIS k p

Remembering that -=* = kt^, while 2^ is on the principal

normal of the evolute.

Theorem 4. A tangent to an analytic curve at a generalpoint will be in the osculating plane at the correspondingpoint of any evolute.

Since (x) lies in the normal plane at {x), we may write

tVXl^ Xi + uii+ VZi,

dXj

ds

—<'i 1 / / ^d/w ti Zi /tj A-\

p

^ du dv

Now T-* is linearly dependent on {x) and (i).

('


and, for the same reason, the assemblage of all terms in (^)and (a) must be a^linear combination of (x) and (x), and soproportional to wWf—Xf = uif+vz^

du p

'd8~ kf uu 1 dp p

f^kda 1

tan.j|j=Jf.c=„.c,,

To get (w) we have(xx) = h^.

w= -^l-^-u' + v'^ Jl+ ^sec2(<T + C).

(xi+ Zi |-) COS (<r + C) + |- f,. sin (<r + C)

Xi= -==== (12)

The coordinates of the point of the line {x) (x) orthogonal

to (oj) will bekXi+ixx:,

>fc^+Mfe^cos(.+C)_ ^„

^^ + C08''(<r+C)

^|^ + C08(«r+(7)

_ J^ + coB^cr + G), ^_.V fc ^ C0s((r + C)

IX ^ ^— > A ^ •

k k

The point in question will therefore have the coordinates

(i sin {(r + C) + Zi cos {(T + C).

COOLIDOE X


This gives us the significance of <r, namely (o- + C) is the fc*

part of the distance from this point to («), i. e. (or+ G) repre-

sents the angle which this normal makes with the principal

normal. If, then, we take two evolutes of our curve the

angle between their corresponding tangents, i. e. those which

meet on the involute, is

Theorem 5. Corresponding tangents to two evolutes of

a curve meet at a constant angle.

Theorem 6. If the generators of a developable surface be

turned through a constant angle about the tangents to one

of their orthogonal trajectories, the resulting surface is

developable.

Theorem 7. The tangents to an evolute of a plane curve

make a constant angle with the plane of the curve.

The foregoing theorems and formulae exhibit sufficiently

the close analogy between the difiFerential theory of curvesin eudidean and in non-euclidean space. It is our next task

to take up the theory of surfaces, and we shall find a noless striking analogy there. We shall mean by an analytic

surface the locus of a point whose coordinates are analytic

functions of two independent parameters. We shall excludefrom consideration all singular points of such surfaces. If

the parameters be {u) and {v), we shall have for the squareddistuice element

ds^ = Edv? + 2Fdudv + Gdv\

\dUdU/^_/7)x'bx

EG-F»=

"hx^ c>a3, Tsx^ SXg

t>U dU ill iu

"iix^ Sa;, "iix^ Sajg

iv iv 3w iv

Q_r'^}x\\3ii iv/'

(13)

This is a positive definite form in the elliptic case, and inthe actual domain of hyperbolic space, to which we shallrestrict ourselves. The discriminant, under this same restric-

tion, will always be greater than zero, for it will vanish onlywhen the tangent plane to the surface is also tangent to theAbsolute.

The equation of the tangent plane at {x) will be

'bx'bxXx^ ^ =0.


The Absolute pole of this plane will be

ix ix

We shall consistently use the letter {y) throughout thepresent chapter to indicate this point. The equation of theplane through the normal, and the point {x+ dx), will be

{Xx) (xx) (xj^)

(Xx) (xx) (a;^)

(^lf)(4-f)^

du + dv = 0.

0..S

i

The cosine of the angle which this plane makes with that

through the normal and the point (x+ bx), or the cosine of

the angle of the two arcs from (x) to (x+ dx) and (x + bx),

will beEdubu + F(dubv+ budv) + Qdvbv

dsbs

The two will be mutually perpendicular if

Edubu+F(dubv + budv) + Gdvbv = 0.

(15)

The condition for perpendicularity between the parameter

curves will be j^^^ ^^g^

The equation of the tangent plane at (x + dx) is

\ X\x { ^ du + ^ dv ) I r- + ^^-idu + ;—^- dv)

"iX "h^X

\'bV ' iuivd'^+^^dv) = 0,

Neglecting differentials of higher order than the first, wehave

XxTiX "iix

û'hV[I

il^XiX Xx "hx S*a;

du Su J)v I]du

Ty^x <>a;

n2

Xx 3 a; "h^x^ « 1 7


The line of intersection with the tangent plane at (x) will be

found by equating to zero separately the first and the last

four terms. This line wOl contain the point (x + hx) if

Ddu bu + iy{du bv+dvtu)+ D"dvhv = 0.

Z) = 3u «)w ivJ^D' =

ix 7>X 3^35

^ ^—Su iv iui>V

iix ^x ()â;

D" = iu i)V ^V''

-/EG-F'(17)

The signs of D, D', D" to be determined presently.

These are the equations for tangents to conjugate systemsof curves, or, briefly put, the equations determining difler-

entials in conjugate directions. The parameter curves will

be mutually conjugate if

D' = 0. (18)

The differential equation for self-conjugate, or asymptoticI1II6B 'Will [)A

Ddw' + 2iydiidv+D"dv^ = 0. (19)

Returning to the point (y), the pole of the tangent plane,

we have

{^) = {ydx) = (xdy) = 0,

/ix 3^\ _ _ / ^\ /^^ _ /^ ^\ _ _ /y _i!^\

Kiv iv) VW '

-{dydx) = Ddu''+Ziydudv+])"dv'^. (20)

These equations will determine the signs of D, If, D".Under what circumstances will the normals at two adjacent

points intersect, i. e. when will their minimum distance bean infinitesimal of higher order than the element of arc?Geometrically we see that the characteristic of the twoadjacent tangent planes must be perpendicular to its conjugate.Conversely, when we do progress iJong such an infinitesimalarc, the tangent plane may be said to rotate about a line


perpendicular to the element of progression, and adjacentnormals are coplanar. At any general point of the surface,

except at an umbilical point where the involution of conjugatetangents is made up of mutually perpendicular tangents,

there will be just two tangents which are mutually conjugateand mutually perpendicular, and these give the elementsdesired.

This fairly plausible geometrical reasoning may easily beput on a sound analytical basis. The necessary and sufficient

condition that the four points (x), (y), (x + dx), (y+dy) shouldbe coplanar is

I

yxdxdy\— 0,

{xx) (xdx) (xdy)

c»a!\ /ix ix

= 0. by (14)

(«lf) (!->«) Q-Jy)

= 0. (21)

Edu + Fdv Ddu + B'dv

Fdu + Gdv D'du + I)"dv

This is the Jacobian of the binary homogeneous forms (13)

and (20), and gives the two tangents which are both mutuallyperpendicular and mutually conjugate; the indeterminatiou

mentioned above occurs in the case where

EiF:G = D:iy:iy'.

Theorem 8. The normals to a surface may be assembled

into two families of developable surfaces. Each normal, with

the exception of those at umbilical points, lies in one surface

of each family.

The integral curves of the differential equation (20) are

called lines of curvature. We see at once that

Theorem 9. If two surfaces intersect along a line which is

a line of curvature for each, they intersect at a constant

angle, and if two surfaces intersect at a constant angle along

& curve which is a line of curvature for one it is a line of

curvature for the other.

This is the theorem of JoachimsthaJ, well known in the

«uclidean case. No less celebrated is the beautiful theorem

of Dupin.

Theorem 10. In any triply orthogonal system of surfaces,

the cui-ves of intersection are lines of curvature.


Let the three families of surfaces be given by the equations

Xi=fi{uv), Xi = <f>i{vw), Xi = fi{wu),

w=*'. (4:)=(«D=('S)=«-As the parameter lines are, in every case, mutually per-

pendicular(7)x ix\ _ /cICB ix\ _ /<>x <>x\ _

/Zx l^x \ /Zx ^x \ _ /1)x a'a; \ /3» ^x \

\iu 'ivZw) V3W JlU 'iv)~ VSu iv 7>W/ \<IV <)W i>u/

_ /c)a; S'â; \ /Zx _^x\ _ _

/ 3x\ _ /ix Sxx _ /<>x <)a;\ _ / S^x Sa3\ _ _

X<)u Su iuhv

= D'VEG-F^ = 0,

The vanishing of 1/ and ^ proves our theorem. Our state-

ment in Chapter XUI that confocal quadrics intersect in lines

of curvature is hereby justified.

A surface all of whose curves are lines of curvature mustbe a sphere. The normal at any point P will determine, withany other point Q of the surface, a plane. The normals to

the surface along this curve, will, by hypothesis, generate anevolute, and hence, by (7) make a fixed angle with the plane

;

and this angle must be null, since, by hypothesis, one normallies in the plane. Hence the normals at P and Q intersect, or

all normals must pass through one point. Evidently theorthogonal surface to a bundle of concurrent lines is a sphere.

Let us suppose that we have a conformal transformation of

space. It will carry a triply orthogonal system of surfaces

into another such system, hence a line of curvature into a line

of curvature. It will, therefore, carry any surface all of

whose curves are lines of curvature into another such surface,

hence

Theorem 11. Every conformal transformation of spacecarries a sphere into a sphere.

Of course a plane is here regarded as a special case of asphere.


Let UB now examine the normals along a line of curvature.

Let r be the distance from the point {x) to the intersection

of the normal there with the adjacent normal, a point whosecoordinates shall be called (x).

dXi dxi r dv; . r V . r rldr

di = di'^n - -£''''% - L^*^'"i -2'»*'°^fcJ^-

Now, by hypothesis, (t-) is linearly dependent on (x)

and (y).

dxi cosj^- dyi sin ^ = A (a;,-+ f^j/^).

But (xdx) = (xdy) = {ydx) = (ydy) = (xy) = 0,

TdXi = dyiiaaT'

}îdu+'plv = i^l&du+ 'J^'d^l.Sit dv kV.ou dv J

In particular, let us take as parameter lines the lines of

curvature

—* = tan^ ^^, ^-^ = tan^^

,

du k iu iv k dv

(dxdy) = dv? +tan-jl tan-r

a; k

{dydy) = -^du'+ -^dv^. (22)

tan^^ tan^^k k

In the general case,

Edu+Fdv= -tanr [Ddu + IXdv],

Fdu + Gdv = - tan T [B'du + D"dv].

Eliminating tan r we get our previous differential equation

for the lines of curvature. On the other hand, if we eliminate

du, dv we get


(Z)D"-i)'2)tan2| +[^D"+ GD-2FI)']tiai'^ + {EG-F') = 0.

(23)

1 1 _ _ ED"+ GD-2FD'

TT^î'^z.* ^2" h{EG-F^] '

k tan -^ k tan -^fc k

1 DD"-D'^

fc2tan-;^tan^ ^ '

(24)

These last two expressions shall be called the mean relative

curvature and the total relative curvature, respectively.

They are, by XL (2), the sum and the product of the curva-

tures of normal sections through the tangents to the lines of

curvature. Notice that they are absolute simultaneous

invariants of the two binary forms (13), (20).

Let us now look at the more general question of the

curvature of a curve on our surface. As, by (4), this does notinvolve derivatives of higher order than the second, the

curvature at any point of a curve of the surface is identical

with that of the curve of intersection of the osculating plane

with the surface. Along our curve u and v will be functions

of 8 the parampCter of length of arc, so that, using our previousnotation,

_ , r^Xi du "iix^ din

^^-"iMld^'^ i^d^yThe cosine of the angle which the principal normal to this

curve makes with the normal to the surface may be written

cos o- = +^

,

— k^

P

ds

(y-\Si _ dtf Xi cos o- _ I

'^s I

j~di'^j' ~y~-\'Wr

j~ Liu^Xda) iuiv ds ds dv'Kds/j

riXf dû Ixi dv^l'^ Id^W^d^ d?y

coBa- _ Ddm,'^ + 2irdudv + D"dir^,

P ~ - k[Edu' + 2Fdudv + Gdv^]

'

The indetermination of sign may be used to make the

curvature essentially positive.


Theorem 12. Meunier's. The curvature of a curve on asurface at any point is equal to the curvature of the normalsection with the same tangent divided by the cosine of theangle which the principal normal makes with the normal tothe surface.

Reverting to our previous expressions r^, r^ and taking thelines of curvature as parameter lines, the curvature of thenormal sections through the tangents to the lines of curvature

1 1

k tan -ri k tan -^k k

dXi = tan^ dyi, hXi = tan -^ 62/j

,

jS=tanÂ <? = tan^D",k k

'^

i tan -J i tan -^

or, if 6 be the angle which the chosen tangent makes with

that to V = cons.

1 cos^d sm'e— = + .

^ k tan -^ k tan -^

Theorem 13. The normal sections of a surface at any point

having the greatest and the least cm-vature are those deter-

mined by the tangents to the lines of curvature.

Theorem 14. If on each tangent to a surface at a point

a distance be laid off equal to the square root of the reciprocal

of the measure of curvature of the normal section with that

tangent, the locus of the points so formed will be a central

conic.

We leave to the reader the task of filling in the details of

the proof of the last theorem, they will come very easily from

considering the equation of a central conic as given in

Chapter XII. Of course the theorem is untrue at a point

where the tangents to the two lines of curvature coincide.

This central conic is called Lupin's Indicatrix in the

euclidean case, and we may well use the same name in

the non-euclidean case also.


The curvature of a surface bears a close relation to the

element of arc of the point (y).

-{dxdy) = Ddu^ + 2D'dudv+D"dv^,

(dydy) = edv? + 2fdudv+ gdv^,

(4D=('g)=^(SlD-^(^:s=''

\^ = iy^\sTy—\-Dr-

,-hy }>ys _ D'Ê+B^G-2DD'FEG-F^

B'iy'E-{DD" + iy'')F+DUOEG-F^

yiy syx _ iy'Ê-2iyiy'F+D'^G

\iiv}>v)~ EG-F'- (edu^+ 2fdudv + gdv^)

1

tan -^ tan -^

\tanj tan^^/

{Edu^ + 2Fdu dv+ Gdv^) +

{Ddv? + 2iydudv + ]y'dv^). (25)

An asymptotic curve has the property that as a point movesalong it, the tangent plane to the surface tends to rotate

about the tangent to tms curve, i. e. the tangent plane to the

surface is the osculating plane to the curve, and the normalto the surface is the binormal to the curve. In dealing withsuch a curve the point {y) on the normal will replace thepoint we previously called (^). The torsion of any asymptoticline will be, by (8),

1 _ "/(dydy)

T kds

But, in the case of an asymptotic curve, the second partof the right-hand side of (25) will be zero, while the paren-


thesis in the first part is equal to ds*, hence, for an asymptoticline , , , ,

(dydy) ^ 1 ^ -1

^W Tîfc^tanjtanj'

It is not difficult to see that the two asymptotic lines at apoint, when real, have torsion with opposite signs, we havebut to look at the special case of a ruled quadric, hence :

Theorem 15. The two asymptotic lines at a point, whenreal, have torsions equal to the two square roots of thenegative of the total relative curvature of the surface.

Theorem 16. In any surface of constant total relative

curvature, the torsion of every asymptotic line is constantand equal to a square root of the total relative curvature, andthe necessary and sufficient condition that a surface shouldhave constant total relative curvature is that the asymptoticlines of one set should have constant torsion. Under these

circumstances the asymptotic lines of the other set will have a

constant torsion equal to the negative of that already given,

and the square of either torsion will be the total relative

curvature.

In speaking of the total curvature of a surface we haveused the word relative. It is now time to explain why that

adjective is chosen. Let us try to express our total relative

curvature in terms of E, F, G and their derivatives. We have

)fc^tanjtanj"^^(^^-^^)" ^ ^

For the sake of simplicity we shall take as parameter lines

u, V the isotropic curves of the surface, i. e. those whosetangents also touch the Absolute. We assume that our

sur&ce is not a developable circumscribed to the Absolute,

and that in the region considered no tangent plane to the

surface touches the Absolute. The isotropic curves at every

point will therefore be distinct. We shall have

E=.Q = 0, (xx) = k\

/ })x\ / 3a;\ / 'i^x\ / 'S^x\ _

2Fdv,dv = d8\

/l^x a^\ _ ^


/OX 0-X\ IF

yiib- Su^/""

\l)u <>v <)u iv.

D'^ =— 1

FF

! -i'

-F

ii^X 'i?X \

nD" = -^ F

F hF()V

iF /S^^

y^kÊG-F^) ~F^lFiu^ ^^v^ ~ k^

1-i. (26)

The first expression on the right is the Gaussian curvature

of a two-dimensional manifold whose squared distance element

is 2Fdudv.*

Theorem 17.t The total relative curvature of a surface is

equal to the diflFerence between its total Gaussian curvature

and the measure of curvature of space.

The Gaussian curvature may also be called the total

absolute curvature. Notice that this theorem remains true

in euclidean space where the measure of curvature is 0.

The problem of finding all surfaces of total relative curva-

ture zero is quickly solved. Let us assume that

rtaa-r = 00-

Then, by an equation just preceding (22), as

Sa;,

t)D:'^0, '£'="•

and there will be the same tangent plane all along u — const.

Theorem 18. A surface of total relative curvature zero is

a developable.

* Cf. Bianchi, loc. cit., p. 68. t Cf. Bianchi, loc. cit., p. 609.

XV DIFFEEENTIAL GEOMETRY 205

Clearly every developable has total relative curvature zero.Much more interest attaches to the surfaces of total Gaussian

curvature zero, i. e. those which are developable upon the

euclidean plane. The total relative curvature wUl be — j^

There is an advantage in considering the hyperbolic andelliptic cases separately.

En the hyperbolic case let (y) be the centre of a sphere, theconstant distance thence to points of the surface being r

cos^=(^), inan^^=Pr^^^^%M^].k K' k \_ {xyY J

If the surface is to be actual (xx) = k^. If the sphere bea proper one {yy) = k^, the total relative curvature will be

> -T2~* -^ *^® "^^^ °^ * horocyclic surface we may not

assume (yy) = k^, but must treat (y) as homogeneous co-

ordinates where (yy) = 0. We get then

1 _ _ 1^

k'^t&n^y^'

k

Theorem 19.* The horocyclic surface of hyperbolic space is

developable on the euclidean plane.

In elliptic space there is a peculiarly notable class ofsurfaces of Gaussian curvature zero, ruled surfaces. We havealready seen one example, the CliflFord Surface of Chapter X.This quadric, be it remembered, cuts the Absolute in twogenerators of each set, and its own generators form an or-

thogonal system. Now Dupin's indicatriz shows that the

normal sections of greatest and of least curvature will be

determined by tangents bisecting the angles of the twogenerators, and the planes of these normal sections will cut

the surface in two circles whose axes are the axes of revolution

of the surface, and whose centres lie on these axes. Thecentres are thus mutually orthogonal points, hence the total

relative curvature is — y^g' ^^^ *^® Gaussian curvature is zero.

This statement was given without proof in Chapter X. Wenotice also that the generators of either set are paratactic,

and the question arises, will not this fact alone constitute

a sufficient condition that a surface should have Gaussian

curvature zero ?

* Cf. Manning, loc. cit., p. 52 ; Killing, Die Grundlagen der Geomelrie, Fader-

born, 1898, p. 33.


Let us imagine that we have a sui-face generated by oo^

paratactic lines.* The parameter v shall give the actual dis-

tance measured on each line from an orthogonal trajectory

V = const. We have for oui- distance element

We know, moreover, by Chapter IX that if two lines be

paratactic they have an infinite number of common per-

pendiculars on which they determine congruent distances.

Hence E is & function of u alone, and we may choose u so

that it shall be equal to unity

ds^=du^ + dv\ (27)

and the Gaussian curvature is zero.

Conversely, suppose that we have a ruled surface of

Gaussian curvature zero. The square of the element of arc

may be writtends^ = Edu^ + dv\

Since the Gaussian curvature is zero

On the other hand we may write our surface parametrically

in the form

«i =fiW cos -^ +<î(u) sin ^

,

with the additional conditions

(//) = {H) = Tc\ iff) = {H') = {/<!>) = im+W) = ;

E={ff)coB'^l +(*>') sin^'l +2(/>')8m|co8|,

kF = (<!>/') cosÎ- {/<!>') sin^ 1 = 0, (/<!>') = (0/') = 0.

These are identical with previous

E = [(9(u)]V+ 2e{u)ylr{u)v + [l/'(u)]^

only when „, . „^ e{u) = 0.

We may, then, take

E=l, d^ = du'^+ dv';

* For an interesting treatment of these surfaces see Bianchl, ' Le superficie

a corratura nulla nella geometria ellitica,' Annali di MaUmatica, Serie 2,

Tomo 24, 1896.


and this shows that two adjacent generators determine equaldistances on all their orthogonal trajectoiies, and so areparatactic.

Theorem 20. The necessary and sufficient condition thata ruled surface in elliptic space should have Gaussian curva-ture zero is that its generators should be paratactic.

Another highly interesting criterion for a surface of constantGaussian curvature zero is obtained as follows

:

E=G=1, F=0;(ix t>*a; \ _ /^x ix\ _ /"bx Sârv _ / S*a3\ _

c>u <>u "bv'~

vJ)!*" Tsv/ ~ \t)u SW ~' v <>atv

The coordinates of the absolute pole of the tangent plane"® ^ Ix Ix

yi =^Si

The coordinates of the absolute pole of the osculating plane

to the orthogonal trajectory of the generatoi-s, i. e. to a curve

V — const., are

>~h.' bx Ty^x

rx 1,

(yO = 0.

This shows that the generators are binormals to their

orthogonal trajectories. Our given surface may be written

in the form

Xi = Xfiu) cos ^ + ii{u) sm ^

.

ds' = dv" + [cos* \ + Yi ^'°'1]

'^^'•

This reduces to

when, and only when

dAi^+dv^,

Theorem, 21. The necessary and sufficient condition that a

ruled surface should have Gaussian curvature zero is that it

should be generated by the binormals to a curve whose

squared torsion is equal to the measure of curvature of space.

The proof given holds equalljr in hyperbolic space ; the

surface is, however, in that case imaginary. If we compare

theorems 16 and 21, we get


Theorem 22. The necessary and sufficient condition that

it should be possible to assemble the normals to a surface into

one parameter families of left (right) paratactics, is that the

given surfifice should have Gaussian curvature zero. It will,

then, be possible to assemble the normals into families of

right (left) paratactics also. The intersections of the given

surface with the various families of paratactics will be the

asymptotic lines of the former.

We shall, as in euclidean space, define as the geodesic

curvature at any point of a curve on our surface, the curvature

of its orthogonal projection on the tangent plane at that point.

Let us denote this by — , while o- is the angle which theP9

,

osculating plane makes with the tangent plane to the surface.

Then, applying Meunier's theorem to the projecting cone

1=5^^ (28)Pg P

As a first exercise, assuming F =0, let us find the geodesic

curvature of one of our parameter lines

cfej = ^/G dv,

. _ k "bXi

P ds k ^G^^'"^'/G î''^ k'

To find cos o- we must determine the distance of (s) fromthe point orthogonal to {x) on the curve v = const., i. e. to the

<r _ 1 /Ix i / 1 3a!\\

P~ s/EG^^^ ^ v^ ^^^ '

(29)pg~ VEG <>«

For the other parameter line

1 -1 Ti's/E

Pg VEG ^V

Let us now, more generally, find the geodesic curvature ofthe curve

, ,


Once more we shall make use of the isotropic parameters,so that E=G = 0,

dvda = y^2Fv'6hi, v'=^,

dv,

*'

>/2Fv'lû iivj'

For an orthogonal trajectory to this curve

by _ dv _ ,

hu du ~ '

hs=hu'/-2Fi/,

— — —

^

X^î _ I ^î\

'^V = i^'*^'

f— '_ ^1 — 1 I 3u dtt

.M + 2?

J

9g ^/-2Fi/l 'iP 2

= 1 r^^.^yaw]. (30)

What will be the nature of those curves whose geodesic

curvature vanishes, i. e. those curves whose osculating planes

pass through the normal? These shall be called geodesic

lines, and, evidently, we shall have

du v'2t7 "^^

This merely tells us that our given curve is an extremal,

i. e. the first variation of the length between two fixed points

is zero. If we assume that two sufficiently near points can

always be connected by a curve of minimum length * weshall get

* For a proof of the existence of this curve, see Bolza, Leckma on Vie Calculus

of rariatima, Chicago, 1904, Ch. VIII.


Theorem 23. The curve of shortest length between two

points of a surface is a geodesic line.

Eemembering 21, we have further

Theorem 24. The orthogonal trajectories of a family of

paratactic lines are geodesies of the surface generated by

these lines.

If we consider the two planes through the normal to a

surface and the two tangents to the lines of curvature, wesee that they are mutually perpendicular, and that each

touches the focal surface of the congruence of normals at the

point of intersection of the two adjacent normals in the other

plane.*

Theorem 25. In any congruence of normals, the edges of

regression of the developable surfaces are geodesies of the

focal surfaces of the congruence.

The osculating plane to any straight line is indeterminate

;

the line is, therefore, a geodesic for all space ; a result also

evident from Chapter II. 30. It is also clear that as the

expressions for the geodesic curvature of a parameter line in

terms of E, F, G and their derivatives are the same in euclidean

and in non-euclidean space, and the formula for the distance

element is written in the same shape, so will the formula for

the geodesic curvature of any curve be the same. We might,

for instance, have given this formula in terms of the Beltramiinvariants. We have, however, purposely avoided the intro-

duction of these into the present work, and wiU thei'efore

merely refer the reader to the current textbooks in differential

geometry.tAs a last problem in the differential geometry of surfaces

let us take up that of minimal surfaces. To begin with, whatwill be the element of area? It is perfectly clear that theexpression for this will be the same as that in the euclideancase. The sine of the angle formed by the parameter lines

will be, by (15)

VEG-F^-/EG

and the area of the elementary quadrilateral

VEG-F^dudv.

* For a simple proof of this general theorem see Picard, loc. cit., vol. i,

pp 307, 808.

-f e. g. Bianchi, DifferenHalgeometrie, cit, p. 253.


Let us, in particular, take the linee of curvature as para-meter lines. The formula for the area enclosed by a givencurve will be

JI'/EGdudv.

Let us compare this with the area enclosed by this curveupon a surface reached by laying off on each normal anextremely small distance w{uv).

— w . wa;i = a;,-cos-^ + 2/,.sm-^,

J— J *". J • '"' 1 r 'w • '"'1

J

axi = aXfCOB— +ayiBm^ — -=- a;^ cos-r^ —^fSm-r law.

The squared element of arc for this surface will be by (22)

i2

d8»=E

. wsm -r

IV kcos-,- +

k , r,

^kdv?+G

. wsm-rw k

cos-r +tan-^

k

dv^-dAJO^

This becomes, when we neglect powers ofw above the first.

dS' = E 1 +

2-k

tan^dii'+G 1 +

For the surface element we have

VEGtan -r + tan -^

2-k

tan^k

dv^

tan -r tan -Â; k

tan -^ tan-j^

dudv.

Developing by the binomial theorem, and neglecting higher

powers of w we have

J -/EG 1 +''tan-,^ + tan^?^

w I k k

tan -J tan -^

o2

dvbdv.


If we define as a minimal surface one where the first

variation of the area is zero, certainly a necessary condition,

we have

Theorem 26. The necessary and sufficient condition that a

surface should be TniTiiTnH.1 is that the mean relative curvature

should be zero.

We see from (23) that the numerator of the expression for

the relative mean curvature is the simultaneous invariant

of (13) and (20), and vanishes when, and only when, the

tangents to the asymptotic lines are harmonically separated

by those to the isob'opic ones, hence

Theorem 27. The necessary and sufficient condition that asurface should be minimal is that the asymptotic lines should

form an orthogonal system.

This theorem justifies our statement in Chapter X that aClifford surface is a minimal surface. It is very interesting

that in non-euclidean space we should have an algebraic

minimal surface (other than the plane) whose order is as lowas two.

We may go one long step further towards the solution ofthe problem of minimal surfaces, namely, exhibit the differ-

ential equations on which they depend.*We shall once more take as parameter lines the isotropic

ones. These will form a conjugate system, since they areharmonically separated by the asymptotic lines, hence

E = G = iy=o,

Tfia-. 1

^,+ p^-.- = 0. (31)

It is merely necessary to find F and take for (x) foursolutions of (3) subject to the restriction (asc) = Jb^.

Let us put

* Cf. Darbouz, Zeporu <ur lo thearie gmirale des mifaea, vol. iii, ch. xir, Paris,

1894. The reader is strongly urged to read this interesting ch^ter in con-nection with the present work.


which is certainly possible, since

"^ au Iv*0.

We easily find

R = Q = 0, FP = ^,a«a!< 1 ZFi)Xi o

Now

Hence ^„ -„

•^^ ^Wso, i) = 0.

The total relative curvature is zero, and the surface is

developable. In a developable surface the asymptotic lines

fall together, by (24); hence a minimal developable must becircumscribed to the Absolute, and cannot be real in the

actual domain. Conversely it is clear that every developablecircumscribed to the Absolute is a minimal surface in that its

asymptotic lines are mutually perpendicular, even though it

lie in a region of our space where the concept area has not

been defined.

In the second case let us suppose 4>{u) ^ 0.

:r= ^) — p • Then

replace the letter u by the letter u once more.

In like mannerf^^l^_F}Xi li

hv^ F i)v 7)v k^'

Multiplying through by y-| and adding

/}^x i^x\ _1^ 1 i?^

214 DIFFERENTIAL GEOMETRY CH. xv

On the other hand

_ / aâ; JZa^ _ j^ ™

l^F _ \-F^ } ^îU(>V ~ k" F hu iv

j^^^'logF^ 1 ^Suit' i*

Lastly, let us put ^ _ ^^^

(32>

ifc2 r—^ + Bin 2m) = 0. (33)

When ^ has been found we may, as already noted, find {x\

from (31).

CHAPTER XVI

DIFFERENTIAL LINE-GEOMETRY

In Chapter IX we gave the foundations of the Fliickerian

line-geometry, and the fundamental invariants of a metrical

character; in Chapter X we saw what advantages arose fromtaking the cross instead of the line as element, and intro-

ducing suitable coordinates. Chapter XY was given to thedifferential geometry of curves and surfaces. It is the object

of the present chapter to draw all of these threads together

into a theory of differential line-geometry, and, in particular,

a theory of two-parameter line systems or congruences.*

We shall define as an analytic line-congruence a systemwhose Fliickerian coordinates are analytic functions of twoindependent parameters, say u and v. This is equivalent to

supposing that our lines are determined by two points, whichwe may assume mutually orthogonal, whose coordinates are

analytic functions of the two independent parameters in

question.

Xi = Xi{v,v), yi = yi{v.v), {xx) = (yy) = k\ (xy) = 0. (1)

Following Rummer's classical method, we shall write the

following fundamental quadratic expression

:

Vo Vi 2/2 2/3I'

= Edu^ + 2Fdudv + Gdv^

k\dxdx)-{ydxf=

k^{dydy)-{xdyf =Xq OOj Xg x^

dy^dyidy^dy^

= Edv? -t- 2F'dudv + Q'dv\ (2)

k\dxdy) - edu^+ (f+f)dudv+gdv\

fix 3ar\ ix\

<:ID-(»3(^S)=^.* The fiist part of the present chapter follows, with slight modifications,

a rather inaccessible memoir by Fibbi, ' I sistemi doppiamente infiniti di

raggi negli spazii di curvatura costante,' Annaii ddla R. Scuola Normale Su-

periare, Fisa, 1891.

216 DIFFERENTIAL LINE-GEOMETRY

^-(^:^:)-('s)'=«

*(^!^D-(4-D=<''

CH.

(3)

(4)

(5)

EG-F^=\yx'bu'bv

= A\ E'G'-F'^ =

(6)

The following relations will subsist between these various

expressions

:

3x/

since

^2 1

^'=^[Ge^-2Ji'e/+^/*].

1

^'=i-. \.Gef'-F{eg+ff') + Efg\A2

<?' = ^,[G/'*-2J'{/sr)+^ff^,

E = ~\G'^-2Fef'+E'f^\

^= ÂGy-neg+fn+E'fgi

G'=ÂGT-2F'(fg)+EYlAA'=(egr-^')-

(7)

(8)

(9)

XVI DIFFERENTIAL LINE-GEOMETRY 217

Notice that A and A' being square roots of positive definiteforma cannot vanish in the real domain.We remember from Chapter IX, that two lines which are

not paratactic have two common perpendiculars meeting themin pairs of mutually orthogonal points. Let us, as a first

problem, find where the common perpendicular to a line ofour congruence and an adjacent line meets the given line.

The coordinates of an arbitrary point of our line may be(T T\xcosj- + 2/Bin-j while an arbitrary point of an

adjacent line vriU be \{x-\-dx)+ix{y+dy).

Let us begin by writing that the second of these points is

orthogonal to \xsmT — y cos^ the point of the first line

orthogonal to the first point, while, on the other hand, the first

point lies in the absolute polar plane of fi{x+dx)—\{y-\-dy).There will result two linear homogeneous equations in A and jii

whose deteiminant must be equated to zero. When this is

simplified in view of the identities

{xdx)= -i (dxdx), (ydy) = -\ (dydy),

(xdy) + (3)dx) = - {dxdy),

we shall have

\le^—\{dxdx)'\ sin ^ — (ydx) cos ^

- [P-^(dy dy)\ sin r - (xdy) cos r

(xdy) sin J -]]e'-l{dydy)'\ cob^k

r= 0. (10)

(ydx) sin jT + [^—^{dxdx)\ cos r

Casting aside infinitesimals above the second order

1^{dxdy) (cos* ^ — sin* A

-{k\dxdx)-{ydxf-k\dydy)-v(xdyf\mi.'^<io&'^ = 0,

(edu* + {f+f)dudv+gd'i^) (cos* ^ - sin*^)

+ [(^- .E'jdtt* + 2(.P-i*)<itMZu+ (ff- G')d«*]sin ^ cos^=0. (11)

218 DIFFERENTIAL LINE-GEOMETRY ch.

This will give oo^ determinations for r in the general case

where

e : (•^) : g ^ {E-E') : {F-F') : {G- G'), (12)

and, as we saw in Chapter X, Theorem 5, with the corre-

sponding eUiptic case, these common perpendiculars will

generate a surface of the fourth order, analogous to the

euclidean cylindroid. We shall call a congruence where

inequality (12) holds a ' general ' congruence.

Let us now ask what are the maximum and minimumvalues for r in (11). Equating to zero the partial derivatives

to du and dv we get

redtt+*^'dul (tan^l - l)

+ {{E- E')du + {F~ F')dv'\iBXiJ= 0,

|^(/±Z)£itt+gdi;] (tan''^ - l)

l{F-F')du-{-{0-G')dv\i&ri'^^ = 0.

Eliminating r we have

\e(F-F') - ''-^^^ {E-E')\dv?

+ \e{G-G')-g{E-E')-\ dudv

+ [if^{Q-G')-g{E~E')\dv^ = 0. (13)

VEach root of this will give two values to tan ^ corresponding

to two mutually orthogonal points. On the other hand, if weeliminate du : dv we get

{eg-Uf+rn (ten*I- l)\[eiG-G')

-iF-F')(f+f) + g{E-E')-](iaji'l - l)tan|

+ [{E-E') (G-G')-(F~Fy] tan'^ = 0. (14)

The left-hand side of this equation is the discriminantof (11) looked upon as an equation in du : dv. It gives,

therefore, those points of the given line where the two per-

pendiculars coalesce. Such points shall be called 'limiting

XVI DIFFERENTIAL UNE-GEOMETRY 219^

pointa'. They will determine two regions (when real) pointby point mutually orthogonal, which contain the intersecbionsof the line with the real common perpendiculars. In thesame way we might find limiting planes through the line

determining two dihedral angles whose faces are, in pairs,

mutually perpendicular, and which when real, with their

verticals, determine all planes wherein lie all real commonperpendiculars to the given line and its immediate neighbours.

Theorem, 1. A line of a Theorem Y. Through ageneral analytic congruence line of a general analytic con-contains four limiting points, gruence wiU pass four limiting

mutually orthogonal in pairs, planes, mutually perpendicu-and these,when real,determine lar in pairs, and these, whentwo real regions of the line real, determine two real re-

where it meets the real com- gions of the axial pencil

mon perpendiculars with ad- through the line which con-

jacent lines of the congruence, tain all planes wherein are

They are also the points where real common perpendiculars

the two perpendiculars coin- to the line and adjacent lines

cide. of the congruence. They are

also the planes in which the

two perpendiculars coincide.

We shall now look more closely into the question of the

reality of limiting points and places. We may so choose ourcoordinate system that the equations of the line in question

shall be 00^ = x^ = 0. Reverting to equation (8) of ChapterX the equation of the ruled quartic surface will be, in the

hyberbolic case

a{-x^+x^XiiD.^+ h{x^-irX^)x^x^ = 0. (15>

Let the reader show * that in the elliptic case we shall have

(Oi-a^ (V+ x^) X1X2+ (% + ag) (a;/+ iBj") XoXs = 0. (15')

To find the limiting points on the line Xi = X2 = 0, equate

to zero the discriminant of this looked upon as an equation in

Xi'.Xg or x^: x.j.

a«(-V+a^'')-46X'a:3' = 0- (16)

{a^-a^''{x^^+ x^y-4>(ft^+a^^x^^x^^^0. (16')

In like manner for the limiting planes we shall have

h\x^

+

xif+4aV*2* = 0- (I'')

{a^â^^{x^^x^)-^{fh.-a^Wî=^- (l?")

* See the author's HwA PryecHve Beomelry, cit., p. 26.


Notice that the centres of gravity of the limiting points

are (1, 0, 0, 0) (0, 0, 0, 1) ; whSe the bisectors of the dihedral

angles of the limiting planes are (0, 1, 0, 0) (0, 0, 1, 0).

If we look more closely into the roots of the last four

equations we see that the roots of (16) are all real, those of

(17) all imaginary. As for the two equations (16') and (17')

the one will have real roots, the other imaginary ones, whence

Theorem 2. In hyperbolic space the limiting points of anactual line are real, and the limiting planes imaginary. Inelliptic space this may occur, or the planes may be all real

and the points all imaginary.

Giving to x^-.x^ one of the values from (16') we see that

V + aâ^ _ ^ 2{a^+ a^XqX^ flj— Ctg

Substituting in (15') we have

a;i + a;^ = or qi^— x^ = 0.

The four limiting points will yield but these two planes,hence

Theorem, 3. The perpen- Theorem 3'. The perpen-diculars at the limiting points diculars in the limiting planesline in two planes called meet the line in two points'principal planes' whose di- called 'principal points 'whosehedral angles have the same centres of gravity are those ofbisectors as pairs of limiting two pairs of limiting points,planes.

Reverting to (16') we see that we may also write

^0 : «3 = ± (vôil v'og) : (7^+ Va^.

Let us pick out a pair of limiting points which are notmutually orthogonal, say

(/oi+ Va^, 0, 0, -/a^- V'^ (-/^H- -/a^, 0, 0, v^- -/a^.

The perpendicular from the point {x) to the line x^ = x^ = <S

meets it in the point {x^, 0, 0, x^. Calling dj, d^ the distancesthence to the limiting points just chosen we have

tan^ = (-/g^- -/^)a!o-(v^+ ^ga;,* (v^+ V^)a!o + ( -/a^- 'â^x^

'

tnn^2_ (•>^-Vâ'o+ (Va^+V^)a;^* -{Va^+Va^Xo+(Vai-Va^)xs'


Further, let (m) be the angle whieh the plane througha?! = 352 = and (a;) makes with the principal plane

a!j + asj = 0.

tan^cos''o) + tan^Bin2o) = 0, (18)

This is, of course, the direct analog of Hamilton's well-known formula for the oylindroid.*

Returning to the notations wherewith we opened thepresent chapter, let us find the focal points of our line, i. e.

the points where it intersects adjacent lines of the congruence,or rather, the points where the distance becomes infinitesimal

to a higher order. Here, if the focal point be

{xcosrj^+ysm-^),

we shall have

T . T ,.7, r + dr , ,.. r+drXf cos^ + yi Bin ^ = (a;, + £?«,•) cos—^ + (y^

+

dyf) sin—^-

dXi cos ^ + dyf sin ^ - ^ (aî sin ^ - y^ cos 'Qdr = 0,

kdr = (xdy),

Multiplying through by ^ and adding, then multiplying

through by -^ and adding again

[edu + fdv] cos -r + [E'du + Fdv] sin r = 0,

[f'du + gdv] cos r + [F'du + G'dv] sin t = 0.

* For the Hamiltonian equation see Bianchi, DifferenHalgeanietrie, cit., p. 261,

For the non-enclidean form here given, cf. Fibbi, loc. cit., p. 67. Fibbi's workis burdened with many long formulae ; one cannot help admiring his skill

in handling such cumbersome expressions at all.

222 DIFFERENTIAL LINE-GEOMETRY CH.

Replacing (ydx) by {—ixdy) we have, similaily

[edu +fdv] sin ^ + [Edu, + Fdv] cos t = 0," *

(19)

[fdu+ gdv] sin t + [Fdu + Gdv] cos r = 0.

Eliminating r

{E'f-Fe)dii? + [E'g-F{f-f)-G'e'\dudv

+ {F'g-G'f)dv^=0,

{Ef-Fe)dv? + [Eg-F{f'-f)-Ge\dudv ^^^^

+ (Fg-Gf)dv'=0.Eliminating du : dv

(E'G' - F'f tan'- i+[E'g- F'{f+f') + G'e] tan|

+ {eg-ff') = 0,

{eg -ff) tan^J+ iEg-F{f+ f) + Ge] tan|

^^^^

Subtracting one of these equations from the other

[(eg-ff)-{E'G'-F'^)] tan^| + [(E-E')g-iF-F) (f+f)

+ {G-G')e]tml+[{EG-F^)-(eg-ff)-] = 0. (22)

We see at once that the middle coefficients ai-e identical in

(14) and (22), and these will vanish when, and only when, weare measuring from a centre of gravity of the roots.

Theorem, 4. The centres of Theorem 4'. The bisectors

gravity of the focal points are of the dihedral angles of twoidentical with those of two focal planes are identical withpairs of limiting points. those of two pairs of limiting

planes.

The focal propei-ties of a congruence of normals are espe-

cially interesting. Here we may suppose that {y) is theAbsolute pole of the tangent plane to the surface describedby (a;). We have then

(^S)=(4:)=(4:)=(4D=«.

-('iS)=/=/'-


Suppose, conversely, that

/= /'•

Let us put x^ = a; J- cos 7; 'r y^saij and show that we may

find r so that our line is normal to the surface traced by (x).

For this it is necessary and sufficient that the point of theline orthogonal to (^) should be orthogonal to every displace-

X sin T — y cos^), we must

haveT T

sin -T (xdx)— cos r (ydx) = 0,

(ydx) = —kdr,

and (ydx) must be an exact differential, i. e.

This condition can be put into a more geometrical form.

Let us, in fact, find the necessary and sufficient condition that

the focal planes should be mutually perpendicular. Writingtheir equations in the form

IXxydx

1= 0,

IXxybx

\= 0,

the numerator of the expression for the cosine of their angle

will be

i* (ybx)

fc* -^bxbx)

(ydx) —^{dxdx) (dxbx)

For perpendicularity,

Edubih+ F{dubv + bv,dv)+Gdvhv = 0.

Now, by (20),

dubu _ Fg-Gf rdu bul _ Ge + F(f'-f)-Egdvbv ~ Ef-Fe ' Idv

"^bv]

~Ef-Fe

Hence^^q_ ^^^ ^_^,^ ^ ^

Let us give the name psewdo-varmal to the absolute polaa*

of a normal congruence. We thus get

= k^ \];?{dxbx)-{ydx) (yhx)\


Theorem 5. The necessary Theorem 5'. The necessary

and sufficient condition that and sufficient condition that a

a congruence should be normal congruence should be pseudo-

is that the focal planes through normal is that the focal points

each line should be mutually on each line should be mu-perpendicnlar. tuaUy orthogonal.

If we subtract one of the equivalent equations (20) from the

other, we get an equation which reduces to (13) when, andonly when

Theorem, 6. The necessary and sufficient condition that

a general congruence should be composed of normals is that

the focal points should coincide with a pair of limiting points.

In a normal congruence let us suppose that (x) traces asurface to which the given lines are normal so that

{ydx) = - (xdy) = 0.

Let us then put

Xi = XiCOB-^ + i/,.sin ^ , ^ = as.-sinj^- y^ cos -

,

where y is constant. We see at once that

{ydx) = - (xdy) = 0.

Tfieorem 7. If a constant distance be laid off on each normalto a surface from the foot, in such a way that the pointson adjacent normals are on the same side of the tangentplane corresponding to either, the locus of the points so foundis a surface with the same normals as the original one.

Let us suppose that we have a normal congruence deter-

mined by mutually orthogonal points (x) and (y), whereXf = Xf(uv) traces a surface, not one of the orthogonal tra-

jectories of the congruence. We shall choose as parameterlines in this surface the isotropic curves, so that

(t>a; ix\ _ /ix })x\ _

The sine of the angle which our given Unc makes with thenormal to this surface is

sinfl = ^(yu)(y'^)

k^(dx aarv


Let all the lines of our congruence be reflected or refractedin this surface in such a way that

sin ^ = 71 sin 0.

We must replace yhyy where

yi=^ny, + K^_, ix Ixte — ;—

It is easily seen that for the new congruence also

/ = /'

Theorem 8. If a normal congruence be subjected to anyfinite number of reflections or refractions, the resulting con-

gruence is normal.

We shall now abandon the general congruence and assumethat, contrary to (IS)

6 :^-^ igîE-E'): {F-F) : (G- G'). (24)

There are two sharply distinct sub-cases which must notbe confused

:

(a)/=/'. (b)/^/'.

In either case, as we readily see, (11) is illusory, and there

is no ruled quartic determined by the common perpendiculars

to a line and its neighbours ; these perpendiculars will either

all meet the given line at one of two mutually orthogonalpoints, or two adjacent lines will be paratactic, and have cxi

'

common perpendiculars.

Our condition for focal points expressed in (23) was inde-

pendent of (12), and this shows that our two sub-cases just

mentioned difler in this, that the first is a normal congruence,

while the second is not. Let [x) be a point where our line

meets a set of perpendiculars, {y) being thus the other such

point. Then under our first hypothesis, we shall have

We see that the focal points will fall into {x) and {y) likewise.

These are mutually orthogonal, and so by equation (26) of the

last Chapter, that the total relative curvature of tiie surface

will be — p or the Gaussian curvature zero. We see also by

theorem (22) of that chapter that it is possible to assemble the

lines of our congruence into families of left or right paratactics

according as we assemble them by means of the one or the

other set of asymptotic lines of the given siu'face. Conversely,


if we have given a congruence of normals to a surface of

Gaussian curvature zero, two normals adjacent to a given one

are paratactic thereunto. There must be, then, two values of

du : dv for which (11), looked upon as an equation in r, becomes

entirely illusory. Hence (24) must hold, and as we have

normal congraence (23) is also true.

We now make the second assumption

We shall still take (x) as a point where the line meets the

various common perpendiculars, so that we may put

We may take as coordinates of a focal plane

«, = j^jtxydx\,

(uit) = k''[Edu^+2Fdudv+Gdi^].

But by (20) this expression vanishes. Hence the focal

planes all touch the Absolute, and the focal surface must bea developable circumscribed thereunto. It is clear that the

lines of such a congruence cannot be assembled into paratactic

families.

This type of congruence shall be called ' isotropic '.*

Let us take an isotropic congruence, or congruence of

normals to a surface of Gaussian curvature zero, and choose(x) and (y) so that

e=Hf+f) = g = 0,

r . rXf = x cos T + 2/ sm T

»

T V(da>dx) = cos^ t (dxdx) + sin^ t (dydy).

* The earliest discussion of these interesting congruences in non-euclideauspace will be found in the author's article ' Les congruences isotropes quiservent a representor les fonctions d'une variable complexe', MH deUa R.Accademia delle Scienze di Torino, zzxiz, 1903, and zl, 1904. In the samenumber of the same journal as the first of these will be found an articleby Bianchi, 'Sulla rappresentazione di Clifford delle congruenze rettilineenello spazio ellitico,' Professor Bianchi uses the word ' isotropic ' to coverboth what we have here defined as isotropic congruences, and also congruencesof normals to surfaces of Gaussian curvature zero, distinguishing the latterby the name of ' normal '. The author, on the other hand, included in hisdefinition of isotropic congruences those which, later, we shall define as' psendo-isotropic '. A discussion of these definitions will be found in a noteat the beginning of the second of the author's articles.


This expression will be unaltered if we change r into —r.Conversely, when such is the case, we must have (dxdy) — 0,

and the congruence will be either isotropic, or composed ofnormals to a surface of Gaussian curvature zero.

Theorem 9. The necessary and sufficient condition that acongruence should be either isotropic, or composed of normalsto a surface of Gaussian curvature zero, is that it should consist

of lines connecting corresponding points of two mutuallyapplicable surfaces, which pairs of points determine alwaysthe same distance. The centres of gravity of these pairs of

points wiU be the points where the vai-ious lines meet the

common perpendiculars to themselves and the adjacent lines.

In elliptic (or spherical) space, there is advantage in study-ing our last two types of congruence from a different point

of view, suggested by the developments of Chapter X.Let us rewrite the equations (11) there given.

{Xoyi-Xiya)-(Xjyj,-Xj,yj) = ^X^. (25)

These equations were originally written under the supposi-

tion that (x) and (y) were homogeneous. At present if we so

choose the unit of measure that A; = 1 we have

UX^X) = (,X,Z) = 1. (26)

These coordinates dX), (^X) were foimerly looked uponas giving the lines through the origin (1, 0, 0, 0) respectively

left and right paratactic to the given line. They may now belooked upon as coordinates of two points of two unit spheres

of euclidean space, called, respectively, the left and right

representing spheres* The representation is not, howevei",

unique. On the one hand the two lines of a cross will be

represented by the same points, on the other, we get the sameline if we replace either representing point by its diameti-ical

opposite. We shall avoid ambiguity by assuming that each

line is doubly overlaid with two opposite ' rays ', meaningthereby a line with a sense or sequence attached to its points,

as indicated in the beginning of Chapter V or end of Chapter

IX. We shall assume that by reversing the signs in one triad

of coordinates we replace our ray by a ray on the absolute

* This representation was first published independently by Study, ' Zurnichteuklidischen etc. ,' and Fubini, ' II parallelismo di Clifford negli spazii

ellitici,' AnncUi detta R. Scmla Normale di Pisa, Vol. ix, 1900. The latter writer

does not, howeyer, distinguish with sufficient clearness between rays andlines.

p 2


polar of its line, while by reversing both sets of signs, wereplace the ray by its opposite.

Theorem 10. There is a perfect one to one correspondence

between the assemblage of all real rays of elliptic or spherical

space, and that of pairs of real points of two euclidean spheres.

Opposite rays of the same line will be represented by dia-

metrically opposite pairs of points, rays on mutually absolute

polar lines by identical points on one sphere and opposite

points of the other. Rays on left (right) paratactic lines will

be represented by identiasd or opposite points of the left (right)

sphere.

Two rays shall be said to be paratactic when their lines are.

Reverting to Theorem 12 of Chapter X.

Theorem 11. The perpendicular distances of the lines of tworays or the angles of these rays are half the difference andhalf the sum of the pairs of spherical distances of their repre-

senting points.

Theorem 12. The necessary and sufficient condition that the

lines of two rays should intersect is that the spherical distances

of the pairs of representing points should be equal ; each will

intersect the absolute polar of the other if these spherical

distances be supplementary.

Theorem 13. Each ray of a common perpendicular to thelines of two rays will be represented by a pair of poles of twogreat circles which connect the pairs of representing points.

It is clear that an analytic congruence may be representedin the form

iXi = iXi{uv), ^Xi = rXi{uv),

or else, in general,

iXi = iXi (^Xi rX^ rX^).

Two adjacent rays will intersect, or intersect one another'spolars if

(diXdiX) = {d,Xd^).

The common perpendicular to two adjacent rays will havecoordinates

The condition that a congruence should be either normal orpseudo-normal is

{diXd^X) = {d^d^),

ihXhiX) = (8,Z8,ar),

xvT DIFFERENTIAL LINE-GEOMETRY 229

= +CX,Z)(,Z8,Z)

I

(jZjZ)(jZfijZ)

\{iXdiX){diXhiX)

from these

{diXhjX) = ± {d^h^). (27)

Let us determine the significance of the double sign. K, in

particular, we take the congruence of normals to a spherewhose centre is (1, 0, 0, 0) we shall get the equations

and this transformation keeps areas invariant in value andsign. On the other hand, the congruence of rays in the

absolute polar of this plane will be

a transformation which changes the signs of all areas. Lastly,

we may pass from one normal congruence to another by a

continuous change, wherein the sign in equation (27) will not

be changed, hence *

Theorem 14. A normal con- Theorem 14'. A pseudo-

gruence will be represented normal congruence will be

by a relation between the two represented by a relation be-

spheres which keeps areas in- tween the two spheres wherevariant in actual value and the sum of corresponding areas

sign, and every such relation on the two is zero, and eveiy

will give a normal congruence, such relation will give apseudo-normal congruence.

Let us next take an isotropic congruence. Here twocommon perpendiculars to two adjacent lines necessarily

intersect, or each intersects the absolute polar of the other.

The same will hold for the absolute polar of an isotropic

congi-uence, a ' psendo-isotropic ' congruence, let us say. Such

a congruence will not have a focal surface at all, but a focal

curve, which lies on the Absolute. On the representing

spheres, in the case of either of these congruences, two inter-

secting arcs of one will make the same angle, in absolute

value, as the corresponding ai'cs on the other. In the par-

ticular case of the isotropic congraence of all lines through

the point (1, 0, 0, 0) the relation between the two representing

spheres is a directly oonformal one, while in the case of the

pseudo-isotropic congruence of all lines in the plane (1, 0, 0, 0)

we have an inversely conformal relation. We may now repeat

* Cf. study, loc. cit., p. 321 ; Fubini, p. 46.


the reasoning by continuity used in the case of the normal

congruence, and get *

Tlieorem 15. The necessaiy Theorem 15'. The neces-

and sufficient condition that a sary and sufficient condition

congruence should be isotropic that a congruence should be

is that the corresponding re- pseudo-isotropic is that the

lation between the represent- corresponding relation be-

ing spheres should be directly tweenthe representing spheres

conformaL should be inversely conformal.

Let us take up the isotropic case more fully. Any directly

conformal relation between the real domains of two euclidean

spheres of radius imity may be represented by an analytic

function of the complex variable. Let us give the coordinates

of points of our representing spheres in the following para-

metric form

:

%u.,—

1

„ z^z.^—\^, -I

» ..A , ^ :r y

' - U,U2-Hl '^ 2 ZiZi+l ^ '

We shall get a real ray when

U2 = Ui, z^ = ij.

In order to have a real directly conformal relation betweenthe two spheres, our transformation must be such as to carrya rectilinear generator into another generator, i. e.

Ml = 1*1 (Zj), 1*2 = Ui(z^). (29)

For an inversely conformal transformation

«1 = '«*l{2^2). «'2= ^li^l)- (30)

All will thus depend on the single analytic function Ui{z).

The opposite of the ray (u) (z) will be

1 , 1Ml = . Zi = ,

^2 2.

1, 1

U., = - —, Z, = .

* First given in the Author's first article on isotropic congruencea, recentlycited.


Let us now inquire under what circumstances the fol-lowing equation will hold

:

uJ--) = ^^^. (31)

If this hold identically, the opposite of every ray of thecongruence will belong thereto. If not, there wiU still becertain rays of the congruence for which it is true. To beginwith it will be satisfied by aU rays of the congruence for

whichu-û^+l = 0, z^z^ + l = 0.

This amounts to putting

0ZjZ) = (,Z,X) = O.

We saw in Chapter X that, interpreted in cross coordinates,

these are the equations which characterize an improper cross

of the second sort, wlucJti 'is macie up ot a pencil ol tangentsto the Absolute. Such a pencil we may also call an improperray of the second sort. Let us see under what circumstancessuch a ray (uz) will intersect a proper ray (uV) oi-thogonally.

Geometrically, we see that either the proper ray must passthrough the vertex of the pencil, or lie in the plane thereof,

and analytically we shall have

(Ui-O («2-0 = («i-2i') (22-0 = 0.

U1U2+ 1 = ZiZi+ 1=0.

There are four solutions to these equations. By considering

a special case we are able to pick out those two where the raylies in the plane of the pencil

It, = Ui, Zj = Zj,

1 1

or else

1 1

tt, = U' Zo = z'

The proper i-ay («.') {s') was supposed to belong to our

congruence. The condition that the improper one (u) (z) shall

also belong thereto will be

(-1^ = --J_


Theorem, 16.* The necessary and sufficient condition that

the opposite of a real ray of an isotropic congruence should

also belong thereunto is that the ray should be coplanar -with

an improper ray of the second sort belonging to the con-

gruence. When the latter ai-e present in infinite number in an

irreducible congruence, the congruence contains the opposite

of each of its rays.

The two cases here given may be still more sharply dis-

tinguished by geometrical considerations. The focal surface

of an isotropic congruence is a developable circumscribed to

the Absolute, and will have a real equation when the con-

gruence is real. There are two distinct possibilities ; first, the

equation of this surface is reducible in the rational domain

;

second, it is not. In the first case the surface is made up of

two conjugate imaginary portions ; in the second there is oneportion which is its own conjugate imaginary. In the first

case there will be a finite number of plajies which touch the

Absolute and also each of the two portions of the focal surface

at the same point, namely, those which touch the Absoluteat the points of intersection of the two curves of contact withthe two portions of the focal surface. In these planes onlyshall we have improper rays of the second sort belonging to

the congruence. If, on the other hand, the focal surface beirreducible, every point of the curve of contact may be lookedupon as being in the intersection of two adjacent planestangent to the Absolute, and the focal surface which is its

own conjugate imaginary. The tangents at each of thesepoints will be improper rays of the second sort of the con-gruence. Theorem 17 may now be given in a better form.

Theorem 17, The necessary and sufficient condition thatan isotropic congruence should contain the opposite of eachof its rays is that the focal surface should be irreducible.

It is very easy to observe the distinction between the twocases in the case of the linear function

az^ + p' yî + s

H $ = —Y, 6 = a, (29) is identically satisfied. But hereit will be seen that if we write

a=a+ bi, y=—c + di,

nought else than the assei

(a, b, c, d). The focal sur

* See the Author's second note on isotropic congruences, p. 13.

our congruence is nought else than the assemblage of all raysthrough the point {a, b, c, d). The focal surface is the cone of

xvt DIFFERENTIAL LINE-GEOMETRY 233

tangents thence to the Absolute, clearly its own conjugateimaginary. On the other hand, when a, fi, y, 8 are not con-nected by these relations, we shall have a line congruence ofthe fourm order, and second class, as is easily verified. It is

well known * that a congruence of the second order and fourthclass has no focal surface, but a focal curve composed of twoconies, so our present congruence has as focal surface twoconjugate imaginary quadric cones which are circumscribedto the Absolute. When their conjugate imaginary centres fall

tog^her in a real point, we revert to the previous case.

When (u) and (s) are connected by the vanishing of apolynomial of order mmu^ and order n in 0j, in the general-

case where (31) does not hold identically, we shall havea line-congruence of order (m + irif. When, however, (31) doeshold, we must subti'act from this the order of the curve of

contact of the focal surface and Absolute, and then divide by2 to allow for the fiact that there are two opposite rays oneach line.

If Ui be a function of Zi that possesses an essential singu-

larity corresponding to a certain value of Zj, we see that as u^takes all possible values (except at most two) in the immediateneighbourhood, there will be & whole bundle of right paratactic

lines in the congruence. If Uj be periodic, there will be aninfinite number of lines of the congruence left paratactic to

each line thereof. If Uj^ be one of the functions of the regular

bodies, we have a congruence which is ti-ansformed into itself

by a group of orthogonal substitutions in (r^), i. e. by a groupof left translations.

We have still to consider the congruence of normals to a

surface of Gaussian curvature zero in ray coordinates. Herethere will be os^ parataotics of each sort to each line. Wemay therefore express (jX) and (,Z) each as functions of oneindependent variable, or merely write

^.(iZ.jZ.jZJ = V^(,Z,,Z,^3) = 0. (32)

All our work here developed for the elliptic case may be

brought into immediate relation with the hyperbolic case, andin so doing we shall get to the inmost kernel of the wholematter. The parameters uÛg will determine generators of

the left representing sphere. They have, however, a moredirect significance. For if u^ remain constant while % varies,

the left paratactics to the ray in question passing through the

point (1, 0, 0, 0) will trace a pencil, and this pencil will lie in

* Cf. Stumi, Oehilde ersler und sweiter Ordnung der Liniengeom^rie, Leipzig,

1892-6, Vol. ii, p. 820.


a plane tangent to the Absolute, for there is only one value

for Uj, namely, , which will make the moving ray tangent•M.2

to the Absolute. When, therefore, u.^ is fixed, one of the left

generatoi-s of the Absolute met by the ray in question is fixed,

and this shows that u-iU^ axe the parameters determining the

left generators which the ray intersects, while z^z^ in like

manner determine the right generators.

If two rays meet the same two generators of one set they

are paratactic, i. e. their lines are. If they meet the same twogenerators of difierent sets, they are either paiallel or pseudo-

paralleL The conditions for parallelism or pseudo-parallelism

will be that two rays shall have the same value for one (u)

and for one (z). Let us, in fact, assume that the subscripts

are assigned to the lettei's uÛj, z-^^^2 in such a way that adirect conformal transformation, or isotropic congruence, is

given by equations (29). Such a congruence wiU contain oo^

rays pseudo-parallel to a given ray, but only a finite numberparallel to it. The conditions for pseudo-parallelism will

thus beitj'= u^, z(= Zi, or nl= u^, Zg = z^. (33)

On the other hand a pseudo-isotropic congruence will begiven by (30), and the conditions for parallelism will be

«,'= Wi, «/= z^, or u.2 = 1*2. %' = ^1- (34)

To pass to the hyperbolic case, let us now assume that

(iX) (^) are two points of the hyperbolic Absolute, and that,

taken in order, they give a ray from (jX) to (,X). Two rayswill be parallel if

(jZ) = (,Z')or(^) = (,Z').

Equations (33) will give the conditions for parataxy, while

(34) give those for pseudo-parallelism. We might push thematter still further by distinguishing between syntaxy andanti-taxy, synparallelism and anti-paraUelism, but we shall

not enter into such questions here. Equations (29) will givea congruence whose rays can be assembled into surfaces withparatactic generators, i. e. a congruence of normals to a surface

of Gaussian curvature zero ; (30) will give an isotropic con-gruence, while (32) will give a pseudo-isotropic congruence.We may tabulate our results as follows.*

* The Author's attention was first called to this remarkable correspondenceby Professor Study in a letter in the summer of 1905. It is developed, withoutproof, but in detail, in his second memoir, ' Ueber nichteukUdische undLiniengeometrie,' Jahresbericht der deitlsclien Mathematikervereinigung, iv, 1906.


Hyperbolic Space.

Ray.Real ray in actual domain, or

pencil of tangents to Abso-lute.

Real parallelism.

Imaginarypseudo-parallelism.Imaginary parataxy.Real congruence of normals to

a surface of Gaussian cur-

vature zero.

Real isotropic congmence.

Real pseudo-isotropic

gruence.

con-

Elliptic Space.

Ray.Real ray.

Real parataxy.

Imaginary parallelism.

Imaginary pseudo-paiallelism.

Real isotropic congruence.

Real pseudo-isotropic con-gruence.

Real congruence of normals to

a surface of Gaussian cur-

vature zero.

CHAPTER XVII

MULTIPLY CONNECTED SPACES

In Chapters I and II we laid do-wn a system of axioms

for om- fundamental objects points and distances, and showed

how, thereby, we might build up the geometry of a restricted

region. We also saw that with the addition of an assumption

concerning the sum of the angles of a single triangle, wewere in a position to develop fully the elliptic, hyperbolic,

or euclidean geometry of the restricted region in question.

Our spaces so defined were not, however, perfect analytic

continua, even in the real domain. To reach such continua

it was necessary to assume that any chosen segment mightbe extended beyond either extremity by a chosen amount.We Saw in the banning of Chapter VII that this assump-tion, though allowable in the euclidean and hyperbolic cases,

will involve a contradiction when added to the assumptions

already made for elliptic space. The diflBculty was oyercomeby assuming the existence of a space which contained as

sub-regions (called consistent regions) spaces where ourprevious axioms held good. For this new type of space weset up our Axionjs I'-VI'.

Our next task was to show that under Axioms I'-V eachpoint will surely have one set of homogeneous coordinates (a;),

and conversely, to each set of real coordinates subject to therestriction that in hyperbolic space

in elliptic space (oex) > 0,

and in euclidean space Xq ^ 0,

there will surely correspond one real point. Under theeuclidean or hyperbolic hypotheses each set of real coor-

dinates can correspond to one real point, at most ; under theelliptic hypothesis, on the contrary, we found it necessaryto distinguish between elliptic space where but one pointgoes with each coordinate set, and the spherical case wheretwo equivalent points necessarily have the same coordinates.One further point was established in connexion with these

developments ; to each point there will correspond but a singleset ofhomogeneous coordinates (x). The proof of this depended

CH. XVII MULTIPLY CONNECTED SPACES 237

upon Axiom VI', which required that a congruent transforma-tion of one consistent region should produce one definite

transformation of space as a whole. Of course such anassumption, when applied to our space of experience, canneither he proved nor disproved empirically. In the presentchapter we shall set ourselves the task of examining whether,under Axioms I'-V of Chapter VII, it be possible to havea space where each point shall correspond to several sets ofcoordinate values.* For simplicity we shall assume that notwo different points can have the same coordinates.

What will be the meaning of the statement that under ourset of axioms two sets of coordinate values (as), (a;') belongto the same point? Let a coordinate system be set up, as

in Chapter Y, in some consistent region; let this region beconnected with the given point by two different sets of over-

lapping consistent regions; then {x) and {of) shall be twodifferent sets of coordinate values lor this point, obtained bytwo different sets of analytic extension of the original coor-

dinate system.

Let us first assume that there is a consistent region whichis reached by each chain of overlapping consistent regions,

a statement which will always hold true when there is a single

point so reached. We may set up a coordinate system in

this region, and then make successive analytic extensions for

the charge of axes from one to another of the overlappingconsistent regions, until we have run through the wholecircuit, and come back to the region in which we started.

If, then, one point of the region have different values for its

coordinates from what it had at the start, the same will betrue of all, or all but a finite number of points of the region,

and the new coordinate values will be obtained from the old

ones (in the non-euclidean cases) by means of an orthogonal

substitution. If (a;) and (aj') be two sets of coordinates for

one point we shall have

0..3

i

Conversely, if these equations hold for any point, they will

represent an identical transformation of the region, and give

two sets of coordinate values for every point of the region.

* The present chapter is in close accord with Killing, Die Orandlagen der

Oeomelrie, Paderborn, 1893, Part iv. Another account will be found in Woods''Forms of Non-Euclidean-Space', published in Lectures on Uathematics, Woods,Van Vleck, and White, New York, 1905.

238 MULTIPLY CONNECTED SPACES ch.

We see also by analytic extension that these equations will

give two sets of coordinate values for every point in space.

There is one possible vaiiation in our axioms which should

be mentioned at this point. It is entirely possible to build

up a geometrical system where IV' holds in general only,

and there are special points, called siTigyZar points, whichcan lie in two consistent regions which have no sub-^region

in common. In two dimensions we have a simple examplein the case of the geometry of the euclidean cone with

a singula!- line. We shall, however, exclude this possibility

by sticking closely to our axioms.

Let us suppose that we have two overlapping systems of

consistent regions going from the one wherein our coordinateaxes were set up to a chosen point P. We may connect 7^

with a chosen point A of the original region by two con-tinuous curves, thus making, in all, a continuous loop. If

now, Pj be a point which will have two diflFerent sets of

coordinate values, according as we arrive at it by the oneor the other set of extensions, we see that our loop is of a sort

which cannot be reduced in size beyond a definite amountwithout losing its chai-acteristic property. This shows that,

in the sense of analysis situs, our space is multiply connected.In speaking of spaces which obey Axioms I'-V , but whereeach point can have several sets of coordinate values, weshall use the term multiply connected spaces.

Suppose that we have a third set of coordinate values fora point of our consistent region. These will be connectedwith the second set by a relation

We see that {x") and {x) are also connected by a relationof this type, hence

Tlicorem 1. The assemblage of aU coordinate transfoi-mationswhich represent the identical transformation of a multiplyconnected space form a group.

If (x) and (x') be two sets of coordinates for the same pointthe expression

(xx')cos"

^/(xx) V(x'x')

cannot sink below a definite minimum value greater thanzero, for then we should have two different points of the same

XVII MULTIPLY CONNECTED SPACES 239

consistent region with the same coordinate values, which wehave seen is impossible (Chapter VII).

For the sake of clearness in our subsequent work let usintroduce, besides our multiply connected space ;Si, a space 2,having the same value for the constant h as our space S,

and giving to each point one set of coordinate values only.

The gi'oup of identical transformations of S will appearin 2 as a group of congruent transfoi-mations, a group whichhas the property that none of its transformations can leave

a real point of the actual domain invariant, nor produce aninfinitesimal transformation of that domain. We lay stress

upon the actual domain of 2, for in S we are interested in

actual points only. Let us further define as fundamentalsuch a region of 2, that every point of 2 has an equivalent

in this region under the congruent sub-group which we are

now considering, yet no two points of a fundamental region

are equivalent to one another. The points of S may be

put into one to one correspondence with those of a funda-

mental region of this sort or of a portion thereof, and,

conversely, such a fundamental region will furnish an exampleof a multiply connected space obeying Axioms I'-V.

Theorem 2, Every real group of congruent transformations

of endidean, hyperbolic, or elliptic space, which carries the

actual domain into itself, and none of whose members leave

an actual point invariant, nor transport such a point aninfinitesimal amount, may be taken as the gi'oup of identical

transformations of a multiply connected space whose points

may be put into one to one correspondence with the points

of a portion of any fundamental domain of the given space

for that group.

Our interest wiU, from now on, centre in the space 2. Weshall also find it advisable to treat the euclidean and the twonon-euclidean cases separately.

We shall begin by asking what groups of congiTient trans-

formations of the euclidean plane fulfil the requirements of

Theorem 2. Every congruent transformation of the euclidean

plane is either a translation or a rotation, but the latter type

is inadmissible for our present purpose. What then are the

groups of translations of the euclidean plane "i The simplest

is evidently composed of the repetitions of a single translation.

If the amplitude of the translation be I, while n is an integer,

positive or negative, this group may be expressed in the form

a/=x+nl, y'=y.

The fundamental regions will be strips bounded by lines


parallel to the y axis, each strip including one of the bounding

lines. A corresponding space S will be furnished by a euclidean

cylinder of circumference I.

What translation groups can be compounded from two given

translations ? It is dear that the lines of motion of the two

should not be parallel. For if, in that case, their amplitudes

were commensurable, we should fall back upon the preceding

system; but if the amplitudes were incommensurable, the

group would contain infinitesimal transformations ; and these

we must exclude. On the other hand, the group compoundedfrom repetitions of two non-parallel translations will suit our

purpose very well. If the amplitudes of the two be I and A,

while m and n are integers, we may write our group in the

The fundamental regions are parallelograms, each including

two adjacent sides, excepting two extremities. The Clifford

surface discussed in Chapters X and XV offers an excellent

example of a multiply connected surface of this type.

It is interesting to notice that with these two examples

we exhaust the possibilities of the euclidean plane. Suppose,

in fact, that P is any point of this plane, that is to say,

any point in the finite domain. The points equivalent to it

under the congruent group in question may not cluster any-where, hence there is one equivalent, or a finite number of

such, nearer to it than any other. If these nearest equivalents

do not all lie on a line with P, we may pick out two of them,non-collinear with P, thus determining one-half of a funda-

mental parallelogram. If the nearest equivalents are collinear

with P (and, hence, two only in number), we may pick out

one of them and one of the next nearest (which will be off

that line, unless we are under our previous first case), andthus construct a parallelogram within which there is noequivalent to P, for every point within such a parallelogramis nearer to one vertex than any two vertices are to oneanother. This parallelogram, including two adjacent sides,

except the vertices which are not common, wUl constitute

a fundamental region, and we are back on the second previouscase. Let the reaHer notice an exactly similar line of reasoningwill show that there cannot exist any single valued continuousfunction of the complex variable which possesses more thantwo independent periods.

In a three-dimensional euclidean space we shall find suitable

groups compounded of one, two, or three independent trans-

lations. The fundamental regions will be respectively layers


between parallel planes, four-faced prismatic spaces, andparaUelopipeds. It is easy to determine how much of thebounding surface should be included in each case. It is alsoevident that there can be no other groups composed oftranslations only, which fulfil the requirements.

Let us glance for a moment at the various forms of straight

line which will exist in a multiply connected euclidean space S,

which corresponds to a euclidean parallelopiped in 2. Thecorresponding lines in 2 shall all pass thi-ough one vertexof the fundamental parallelopiped. If the line in S be oneedge of the parallelopiped, the line in iS will be a simple loopof length equal to one period. If the line in 2 connect the

vertex with any other equivalent point, the line in S will still

be a loop, but of greater length. If, lastly, the line in 2 donot contain any other point equivalent to the vertex, the line

in S win be open, but, if followed sufficiently far, will pass

again as close as desired to the chosen point.

There are other groups of motions of euclidean space,

besides translations which give rise to multiply connectedspaces. An obvious example is furnished by the repetitions

of a single screw motion. This may be expressed, n being

an integer, in the form

of= xooanB—y ain n9, j/^XBinnO+ ycosnO, z'=z+nd.The fundamental regions in 2 will be layers bounded by

parallel planes. In <S we shall have various types of straight

lines. The Z axis will be a simple closed loop of length d.

Will there be any other closed lines in S'i The corresponding

lines in 2 must be parallel to the axis, there being an infinite

number of points of each at the same distance from that axis.

When d and 2ir ai-e commensurable, we see that every parallel

to the Z axis will go into a closed line of the type required,

when and 2ir are iiicommensurable, the Z axis is the only

closed line.

Let us now take two points of 2 separated by a distance r

^= sB + rcosa,

rj = y + rcos/3,

(= z + r cos y.

The necessary and sufficient condition that they should be

equivalent is xeoand—yainnO = x+ r cos a,

xamne+ycoane=:y + rooafi,

nd = r cos y.

The last of these equations shows that a line in 2 per-


pendicular to the Z axis (i.e. parallel to a line meeting it

perpendicularly) cannot return to itself. On the other hand, if

eosa = cosy3 = 0; ft0 = 2m7r,

and we have a closed loop of the type just discussed. If

a, /3, y, Ti, be given, r may be determined by the last equation,

and X, y from the two preceding, since the determinant of

the coefficients will not, in general, vanish. We thus see that

in S the lines with direction angles a, /3, y, and possessing

double points, will form an infinite discontinuous assemblage.

If, on the other hand, x, y, z, n be given, a, /3, y, r may be

determined from the given equations, coupled with the fact

that the sum of the squares of the direction cosines is unity

;

through each point in S, not on the Z axis, will pass an infinite

number of straight lines, having this as a double point.

The planes in S will be of three sorts. Those which aie

perpendicular to the Z axis will contain open lines only, those

whose equations lack the Z term will contain all sorts of lines.

Other planes will contain no lines which are simple loops.

Another type of multiply connected space will be deter-

mined by x'={-Vfx+ ma,

y'={-\fy + nh,

z'= z + lc,

I, m, n being integers.

The fundamental regions in S will be triangular rightprisms. Lines in 2 pai-aUel to the Z axis will appear in Sas simple closed loops of length 2c. To find lines which cross

themselves, let us write

a+rcosa = (—l)'a! + ma,

y + reosp = (— lYy+ nb,

z+r cos y = z + lc.

For each even integral value of I, and each integral valueofm and n, we get a bundle of loop lines in S with directioncosines ^„

cos a — &c.

When I is odd, we shall have through each point an infinitenumber of lines which have a double point there, the directioncosines being

— 2x + macos a =

_ _ , &o.V{-2x + 7naf+{-ily + nbY + l^c^

Such lines will, in general, be open. We see, however, that


whereas the length of a loop perpendicular to the x, y plane

is 2c, if the point -g- , — happen to be on such a loop, this

point is reached again after a distance C. This loop has,

therefore, the general form of a lemniscate.*When -we turn from the euclidean to the hyperbolic

hypothesis, we find a less satisfactory state of affairs. Thereal congruent group of the hyperbolic plane was shown inChapter VIII to depend upon the real binary group

the homogeneous coordinates (t) being supposed to define

a point of the absolute conic. The two &ed points mustbe real, in order that the line joining them shall be actual,

and its pole, the fixed point, ideal, m other words, we wishfor groups of binaiy linear substitutions which contain

members of the hyperbolic type exclusively. Apparentlysuch groups have not, as yet, been found. It might seem,

at first, that parabolic transformations where the two fixed

points of the conic fall together, would also answer, butsuch is not the case. We may show, in fact, that in sucha substitution there wiU be points of the plane which are

transformed by as small a distance as we please. The pathcurves are horocycles touching the absolute conic at the fixed

point : having in fact, four-point contact with it. It is mei'ely

necessary to show that a horocycle of the family may be foundwhich cuts two lines through the fixed point in two points

a,s near together as we please. Let this fixed point be (0, 0, 1)while the absolute conic has an equation of the form

^0^ + x^x.^ = 0.

The general type for the equation of a horocycle tangent

at (0, 0, 1) will be

{Xo' + XiX2)+pXi^ = 0.

This will intersect the two lines

Xf,—lxi=:0, x^—mxi = 0,

in the points (l, l,-(P+p)) (m, 1,— (m^+^j)). The cosine of

the kth part of their distance will be

(1-70)" + 2p2p

* These and the preceding example are taken from Killing, Giundlagen,

lots. cit. The last is not, however, worked out.

Q2

244 MULTIPLY CONNECTED SPACES CH.

an expression which will approach unity as a limit, as -

approaches zero."

The group of hyperbolic motions in three dimensions wiU,

as we saw in Chapter VIII, depend upon the linear function

of the complex vaiiable az + B

yz + h

The group which we require must not contain rotations

about a line tangent to the Absolute, for the reason whichwe have just seen, hence the complex substitution must not

be parabolic. Again, we may not have rotations about actual

lines, hence the path curves on the Absolute may not be conies

in planes through an ideal line (the absolute polar of the axis

of rotation) ; the substitutions may not be elliptic. The onlyallowable motions of hyperbolic space are rotations aboutideal lines, which give hyperbolic substitutions, and screwmotions, which give loxodromic ones. There does not seemto be any general theory of groups of linear transformations

of the complex variable, which include merely hyperbolicand loxodromic members only.*The group of repetitions of a single rotation about an ideal

line may be put into the form (Je^ = — 1),

x^= ij cosh nd — x^Bmnd,

K= *i.

asj'= Ao sinh fl + ij cosh 9.

The fundamental regions in S will be bounded by pairsof planes through the line

•''0 ^^ ""s ^ 0"

The orthogonal trajectories of planes through this line willbe equidistant curves whose centres lie thereon. A line in 2connecting two points which are equivalent under the groupwiU appear in /S^ as a line crossing itself once.We may, in like manner, write the group of repetitions

of a single screw motion

*o'= ^0 ^'osh nB—x^ sinh nQ,

x-[= a!i cos ii^—iBj sin 71.0,

x^= Xy sin 71,0+ isj cos 710,

x^=. Xg sinh nd + x^ cosh nd.* For the general theory of discontinuous groups of linear substitutions,

see Fricke-KIein, Vorlesungen uber die Tkeorie der autmiorphen Funktionen, vol. i,

Leipzig, 1897,

xvii MULTIPLY CONNECTED SPACES 245

In elliptic space we obtain rather more satisfactory results.

Every congruent transformation of the real elliptic plane is

a rotation about an actual point, there being no ideal points.

Hence, there are no two-dimensional multiply connectedelliptic spaces. In three dimensions the case is different Letus assume that A; = 1, and consider the group of repetitions

of a single screw motion. The angle of rotation about oneaxis is equal to the distance of translation along the other,

and the two distances or angles of rotation must be of the

form — > ^ in order that there shall be no infinitesimalV V

transformations in the group. Moreover, these two fractions

must have the same denominator, for otherwise the groupwould contain rotations. We may therefore write the general

equations

Xa = a?„cos7i X, sm 71— f

V V

, . Kit AirX, = ajnSinii— +a;, cosTi— >

* " V V

, UTT . ttir

Xn = a;, COB 71 x^smn— j

, . UTI fJiTT

X, =x„ainn— +X3C0S'Ji— »

where A, fi, v are constant integers, and n a vaiiable integer.

It will be found that the cosine of the distance of the points

(x), (a/) will be equal to unity only when n is divisible by v,

i. e. we have the identical transformation, so that there are noreal fixed points nor points moved an infinitesimal distance.

If A = ju we have a translation (cf. Chapter VIII), for our

transformation may be written in the quaternion form:*

{x^ + xî + x^j + x^k)

(Air . Att .\ , . ,

.

cos 71 +Bin7l %\(Xf^-\-Xjl-{-X23+X^K).

The path-curves in 2 will be lines paratactic to either axis

of rotation, and they will appear in <S as simple closed loops

of length - . Notice the close analogy of this case to the

simplest case in eudideain space.

* Killing, Gnmdiagm, cit. p. 342, erroneously states that these translations

are the only motions along one fixed line yielding a group of the desired

type. The mistake is corrected by Woods, loc. cit., p. 68.

246 MULTIPLY CONNECTED SPACES CH. xvii

There is another translation group of elliptic space giving

rise to a multiply connected space of a simple and interesting

description. Let Aj:A2 be homogeneous parameters, locating

the generators of one set on the Absolute. Each linear trans-

formation of these will determine a translation, hi particular,

if we put Xa + ixi = Xj, x^-ix^ = A^,

then the translation

{Xa' + Xi'i + x^j + Xsk) = {a+ bi + cj+dk){Xg + Xjî + xJ + X3k),

may also be written

Xj'= (a + bi)X.i—{c + di) k^,

Xg'= (c— di) Xj + (a— bi) Xg.

Now this is precisely the formula for the rotation of the

euclidean sphere. The cosine of the distance traversed bythe point (x) will be

Va'^ + b' + c^ + d''

which becomes equal to unity only when 6 = c = d = 0, i.e.

when we have the identical transformation. The groups of

elliptic translations which contain no infinitesimal trans-

formations, are therefore identical with those of euclidean

rotations about a fixed point which contain no infinitesimal

members, whence

Theorem 3.* If a multiply connected elliptic space betransformed identically by a group of translations, that group

is isomorphic with one of the groups of the regular solids.

Conversely each group of the regular solids gives rise to agroup of right or left elliptic translations, suitable to define

a multiply connected space of elliptic type.

Of course the inner reason for this identity is that a real

line meets the elliptic Absolute in conjugate imaginary points,

corresponding to diametral imaginary values of the parameterfor either set of generators, and a real point of a euclidean

sphere is given by the value of its coordinate as a point of

the Gauss sphere, while diametrically opposite points will begiven by diametral values of the complex variable. Theproblem of finding elliptic translations, or euclidean rotations,

depend therefore, merely on the problem of finding linear

transformations of the complex variable which transport

diametral values into diametral values.

* Cf. Woods, loc. cit., p. 68.

CHAPTEE XVIII

THE PROJECTIVE BASIS OF NON-EUCLIDEANGEOMETRY

Our non-euclidean system of metrics, as developed in

Chapter YU and subsequently, rests in the last analysis,

upon a projective concept, namely, the cross ratio. The groupof congruent transformations appeared in Chapter YII as

a six-parameter collineation group, which left invariant acertain quadric called the Absolute. An exception must bemade in the eudidean case where the congruent group wasa six-parameter sub-group of the seven-parameter group whichleft a conic in place. We thus come naturally to the idea

that a basis for our whole edifice may be found in projective

geometry, and that non-euclidean metrical geometry may be

built up by positing the Absolute, and defining distance as

in Chapter YII. It is the object of the present chapter to

show precisely how this may be done, starting once moreat the very beginning.*

Axiom I. There exists a class of objects, containing at

least two distinct members, called points.

Axiom II. Each pair of distinct points belongs to a single

sub-class called a line.

The points shall also be said to be on the line, the line

to pass through the points. A point common to two lines

shall be called their intersection. It is evident from Axiom II

that two lines with two common points are identical. Wehave thus ruled out the possibility of building up spherical

geometry upon the present basis.

Axiom III. Two distinct points determine among the

remaining points of their line two mutually exclusive sub-

classes, neither of which is empty.

If the given points be A and B, two points belonging to

* The first writer to set up a suitable set of axioms for projective geometry

was Fieri, in his Principii deUa geometria di posiziojie, oit. He has had manysuccessors, as Enriques, Lenioni di geometria proiettiva, Bologna, 1898, or Vahlen,

Abstrakie Geometrie, cit., Parts II and III. Veblen and Young, ' A system of

axioms for projective geometry,' American Journal of Mathematics, Vol. xxx,

1908.

248 THE PROJECTIVE BASIS OF CH.

different classes according to Axiom III shall be said to be

separated by them, two belonging to the same class wAseparated* We shall call such classes separation classes.

Axiom IV. If P and Q be separated by A and B, then

Q and P are separated by A and B.

Axiom V. If P and Q be separated by A and B, then

A and B are separated by P and Q.

We shall write this relation PQ AB or AB PQ. If PQ

be not separated by A and B, though on a line, or collinear,

with them, we shall write PQ{aB.

Axiom VI. if four distinct collinear points be given there

is a single way in which they may be divided into twomutually separating pairs.

Theorem 1. AB {cD and AE ^GD, then EB-icD.

For C and D determine but two separation classes on the

line, and both B and E belong to that class which does not

include A.

Theorem 2. If five collinear points be given, a chosen pair

of them will either separate two of the pairs formed by the

other three or none of them.

Let the five points heA,B, 0, D, E. Let AC \DE. Then, if

BC {be, AbIdE, and if AB ^DE, BciDE. But if we had

BCÊ and ABî^, ABC would belong to the same

separation class with regard to DE, and hence AC-LDE.

Theorems. If Ac\bD and AE^CD, then AE^BD.

To begin with Bcj.AD, ECiAD; hence BEA.AD. Again,

if we had AB [eD, we should have AB [eC, i.e. AeIbC.

But we have AE CD, hence AE BD a contradiction with

* The axioms of separation were first given by Yailati, 'SuUe proprietacaratteristiche delle variety a una dimensioue,' Riviata di Matanatica, T, 1896.

xvin NON-EUCLIDEAN GEOMETRY 249

AB\eD. As a result, since BeLaD and AbIeD, we must

have AE\BD.

It will be dear that this theorem includes as a special caseTheorem 3 of Chapter I. We have but to take A &t & greatdistance.

Th&yrem 4. If PA fcZ), PB^CD, PQ (aB, then PQ fcZ).

The proof is left to the reader.It will follow from the fact that neither of our separation

classes is empty that the assemblage of all points of a lineis infinite and dense. We have but to choose one point ofthe line, and say that a point is between two others whenit be separated thereby from the chosen point.

Axiom Vn. if all points of either separation class deter-mined by two points A,B,he so divided into two sub-classesthat no point of the first is separated from A by B anda i>oiiit of the second, there will exist a single point C ofthis separation class of such a nature that no point of thefirst sub-class is separated from A by B and C, and noneof the second is separated from B by A and C.

It is clear that G may be reckoned as belonging to either

sub-class, but that no other point enjoys this property.This axiom is one of continuity, let the reader make a careful

comparison with XVIII of Chapter II.

Axiom YIII. All points do not belong to one line.

Definition. The assemblage of all points of all lines deter-

mined by a given point and all points of a line not containing

the first shaJl be called a plans. Points or lines in the sameplane shall be called coplanar.

Axiom IX. A line intersecting in distinct points two of

the three lines determined by three non-collinear points,

intersects the third line.

Let the reader compare this with the weaker Axiom XVIof Chapter I.

Theorem, 5. A plane will contain completely every line

whereof it contains two points.

Let the plane be determined by the point A and the line

BC. If the two given points of the given line belong to BCor be A and a point of BC, the theorem is immediate. If not.


let the line contain the points R and C of AB and AGrespectively. Let P be any other point of the given line.

Then BP will intersect AC, hence AP will intersect BC or

will lie in the given plane.

Thewem, 6. \i A, B, C be three non-collinear points, then

the planes determined by A and BG, by B and GA, and byG and AB are identical.

We have but to notice that the lines generating each plane

lie wholly in each of the others.

Theorem 7. If A', B', C' be three non-collinear points of the

plane determined by ABG, then the planes determined byA'B'G' and ABG are identical.

This will come immediately from the two preceding.

Theorem, 8. Two lines i» the same plane always intersect.

Let B and G be two points of the one line, and A a point

of the other. If A be also a point of BG the theorem is proved.

If not, we may use the point A and the line BG to determinethe plane, and our second line must be identical with a line

through A meeting BG.

Axiom X. All points do not lie in one plane.

Definition. The assemblage of all points of all lines whichare determined by a chosen point, and all points of a planenot containing the first point shall be called a apace.

We leave to the reader the proofs of the following verysimple theorems.

Theorem 9. A space contains completely every line whereofit contains two points.

Theorem 10. A space contains completely every planewhereof it contains three non-coUinear points.

Theorem. 11. The space determined by a point A and theplane BGD is identical with that determined by B and theplane GDA.

Theorem 12. If A', B', G", D' be four non-coplanar points ofthe space determined by A, B, G, D, then the two spaces deter-mined by the two sets of four points are identical.

With regard to the last theorem it is clear that aU points ofthe space determined by A', R, C, ZX lie in that determined byA, B, G, D. Let us assume that R, C', D' are points of AB, AG,AD respectively. The planes BCD and RC'D^ have a common

xviii NON-EUCLIDEAN GEOMETRY 251

line I, which naturally belongs to both spaces. Let us first

assume that AA' does not intersect this line. Let A" be theintersection of AA' with BCD. Then A"B meets both A'B'and Z, hence, has two points in each space, or lies in each.

Then the plane BCD lies in both spaces, as do the line A'A"and the point A ; the two spaces are identical. If, on theother hand, AA' meet I in A", then A lies in both spaces.

Furthermore A'B will meet A"B' in a point of both spaces,

80 that B will lie in both, and, by similar reasoning, C and Dlie in both.

Theorem. 13. Two planes in the same space have a commonline.

Theorem 14. Three planes in the same space have a commonline or a common point.

Practical Ivmitation. All points, lines, and planes herein-

after considered are supposed to belong to one space.

Theorem 15. If three lines AA', BB', GO' be concurrent,

then the intersections of AB and A'R, of BC and B'C, of CAand CA' are collinear, and conversely.

This is Desargues' theorem of two triangles. The following

is the usual proof. To begin with, let us suppose that the

planes ABC and A'B'C are distinct. The lines AA', BB',and CC will be concurrent in outside of both planes. Thenas AB and A'B' are coplanar, they intersect in a point whichmust lie on the line Z of intersection of the two planes ABGand A'B'C', and a similar remark applies to the intersections

of BG and B'C', of GA and G'A'. Conversely, when these

last-named three pairs of lines intersect, the intersections

must be on I. Considering the lines AA', BB', and GG', wesee that each two are coplanai*, and must intersect, but aUthree are not coplanar. Hence the three are concurrent.

The second case occurs where A'B'G' are three non-collinear

points of the plane determined by ABG. Let V and V betwo points without this plane collinear with the point of

concurrence of AA', BB\ GG'. Then VA will meet V'A' in

A", VB will meet V'B' in B", and VC will meet V'G' in G".

The planes ABC and A"B"G" will meet in a line I, andJ5"(7'^will meet both BG and B'C' in a point of I. In the

same way GA will meet G'A' on I, and AB will meet A'B'

on I. Conversely, if the last-named three pairs of lines meet

in points of a fine I in their plane, we may find A"B"G"non-collinear points in another plane through I, so that B"G"meets BC and B'C' in a point of I, and similarly for G"A"

,

252 THE PROJECTIVE BASIS OF ch.

CA, G'A' and for A"B", AB, A'B'. Then by the converse

of the first part of our theorem AA", BR', CO" will be

concurrent in V, and A'A", B'B", CO" concurrent in V.Lastly, the three coaxal planes W'A", VV'B", VV'C" -will

meet the plane ABC in three concurrent lines AA', BB' , CO'.

We have already remarked in Chapter VI on the dependence

of this theorem for the plane either on the assumption of the

existence of a third dimension, or of a congruent group.

Definition. If four coplanar points, no three of which are

collineai', be given, the figure formed by the three pairs of

lines determined by them is called a complete quadrangle.

The original points are called the vertices, the pairs of lines

the aides. Two sides which do not contain a common vertex

shall be said to be opposite. The intersections of pairs of

opposite sides shall be called diagonal points.

Theorem, 16. If two complete quadrangles be so situated

that five sides of one meet five sides of the other in points

of a line, the sixth side of the first meets the sixth side of the

second in a point of that line.

The figure formed by four coplanar lines, no three of whichare concurrent, shall be called a complete quadrilateral.

Their six intersections shall be called the vertices ; two vertices

being said to be opposite when they are not on the same side.

The three lines which connect opposite pairs of vertices shall

be called diagonals.

Definition. If A and G be two opposite vertices of a com-plete quadrilateral, while the diagonal which connects themmeets the other two in B and D, then A and B shall be said

to be harmonically separaied by C and D.

Theorem, 17. If A and C be harmonically separated byB and D, then B and D are harmonically separated by Aand C.

The proof will come immediately from 15, after drawingtwo or three lines; we leave the details to the reader.

D^nition. If A and C be harmonically separated byB and D, each is said to be the harmonic conjugate of theother with regard to these two points ; the four points mayalso be said to form a harmonic set.

Theorem, 18. A given point has a unique harmonic conjugatewith regard to any two points collinear with it.


XVIII NON-EUCLIDEAN GEOMETRY 253

Theorem 19. If a point be connected with four pointsA, B, C, D not collinear with it bylines OA, OB, OC, OD, andif these lines meet another line in A', B', &, D' respectively,

and, lastly, if A and C be harmonic conjugates with regardto B and D, then A' and C are harmonic conjugates withregard to B' and If.

We may legitimately assume that the quadrilateral con-

struction which yielded A, B, G, D was in a plane which didnot contain 0, for this construction may be effected in anyplane which contains AD. Then radiating lines throughwill transfer this quadrilateral construction into anothergiving A', B', C", ly.

D^niticm. If a, b, c, d be four concurrent lines which pass

through A,B,C,D respectively, and HA and C be harmonically

separated by B and D, then a and c may properly be said

to be harmonically separated by h and d, and h and dharmonically separated by a and c. We may also speak of

a and c as harmonic conjugates with regard to h and d, or

say that the four lines form a harmonic set.

Thewem 20. If four planes a, ;3, y, 8 determined by a line I

and four points A, B, G, D meet another line in four points

A', W, G', D' respectively, and if A and G be harmonically

separated by B and D, then A' and C are harmonically

separated by B' and 2)'.

It is sufficient to draw the line AD' and apply 19.

Definition. If four coaxal planes a, /3, y, 8 pass respectively

through four points A, B, G, D where A and G are harmonically

separated by B and D; then we may speak of a and y as

hormonicaUy separated by ^ and 8, or /3 and 8 as harmonically

separated by a and y. We shall also say that a and y are

luumonic conjugates with regard to /3 and 8, or that the four

planes form a harmonic set.

We shall understand by projection the transformation

(recently used) whereby coplanar points and lines are carried,

by means of concurrent lines, into other coplanar points and

lines. With this in mind, we have the theorem.

Theorem 21. Any finite number of projections and inter-

sections will carry a harmonic set into a harmonic set.

Axiom XI. If four coaxal planes meet two lines respec-

tively in A, B, G, D and A', B', C', U distinct points, and

if AG^BD then A'C'h'D'.


Definition. If AC BD and I be any line not intersecting

AD, we shall say that the planes IA and IC separate the

planes IB and ID.

Definition. If the planes a and y separate the planes )3

and 6, and if a fifth plane meet the four in a, h, c, d respec-

tively, then we shall say that a and c separate b and d.

A complete justification for this terminology wiU be found

in Axiom XI and in the two theorems which now follow.

Theorem 22. The laws of separation laid down for points

in Axioms III-VII hold equally for coplanar concurrent lines,

and coaxal planes.

We have merely to bring the four lines or planes to intersect

another line in distinct points, and apply XI.

Theorem 23. The relation of separation is unaltered by anyfinite number of projections and intersections.

Theorem 24. If ^, B,C,D be four collinear points, and A

and C be harmonically separated by B and D, then AC \ BD.

We have merely to observe that our quadrilateral con-struction for harmonic separation permits us to pass bytwo projections from A, B, C, D to C, B, A, D respectively, so

that if we had AB CD we should also have CB AD, and

vice versa. Hence our theorem.

Before proceeding further, let us glance for a moment at thequestion of the independence of our axioms.The author is not familiar with any system of projective

geometry where XI is lacking. X naturally fails in planegeometry. Here IX must be suitably modified, and Desargues'theorem, our 15, must be assumed as an axiom. IX is mak-ing in the projective euclidean geometry where the ideal

plane is excluded. VlII fails in the geometry of the single

line, whUe VH is untrue in the system of all points withrational Cartesian coordinates. IH, IV, V, VI may be shownto be serially independent.* II is lacking in the geometryof four points.

Besides being independent, our axioms possess the far moreimportant characteristic of being consistent. They will besatisfied by any class of objects in one to one correspon-

* Vailati, loc. oit., note quoting Padoa.


dence with all sets of real homogeneous coordinate valuesXq-.xîx^-.x^ not all simultaneously zero. A line may bedefined as the assemblage of all objects whose coordinatesare linearly dependent on those of two. If A and C havethe coordinates (x) any (y) respectively, while B and D havethe coordinates k(x) + iJi(y) and \'(x) + ii{y), then A and Cshall be said to be separated by B and D if

When this is not the case, they shall be said to be notseparated by B and 2).

As a next step in our development of the science of pro-

jective geometry, let us take up the concept of cross ratio.

Suppose that we have three distinct collinear points I^, I^, !{.

Construct the harmonic conjugate of i^ with regard to P,

and i^, and call it F^, that of I{ with regard to I^ and J^,and call it F^, that of ij with regard to ij and i^, andcall it Pi, and so, in general, construct i^+j and i^_iharmonic conjugates with regard to i^ and i^. The con-

struction is very rapidly performed as follows. Take and Vcollinear with P^, while our given points lie on the line Iq.

Let ^1 be the line from the intersection of OP^ and VPg to P^

.

Then OI^+i and VI^ will always intersect on Zj, the generic

name for such a point being Q„+j.*

Theorem 25. .^.^+1 [.^P, if % > 0.

The theorem certainly holds when n = 1. Suppose that

Po-P„J-Pn-i-P«- We also know that P„-iP„+iJP„P„. Hence,

clearly PoPn+^^Pn^^. We notice also that Po.^+2jp„P,,

and, in general I^ I^+jc \^P^. A similar proof may be found

for the case where negative subscripts are involved.

Theorem 26. If P be any point which satisfies the condition

ijP L^ J^ , then such a positive integer n may be found

that PoPJjPn^:^, -Po-ZJI+iJ-P-Poo-

Let us divide all points of the separation class determined

by .^ i^ which include ij and P the positive separation class

let us say, into two sub-classes as follows. A point A shall

be assigned to the first class if we may find such a positive

* See Fig. 4 on page following.

256 THE PROJECTIVE BASIS OF OH.

integer n that ij i^+j AI^ , otherwise it shall be assigned

to the second class, i.e. for every point of the second class

and every positive integral value of n, ^B i^+ii^- Then,

by 3, as long as A and B are distinct we shall have

^^Ldi^, giving a dichotomy of the sort demanded by

Axiom VII, and a point of division D. Let us further assume

that OB meets l^ in B, and FFmeets l^ in C. We know that

îWoQw Hence lines from ij to V and ^D are not

separated by those to and i^. Hence lines from D to Pq

and V, are not separated by those to and I^, so that

Poo

Fio. 4.

I^C\.BP^ or C is a point of the first sub-class. We may,

then, find n so great that P„P„\GP^, hence QiQ„+i { BP^

and P^P„^,JBP^. But P^P^JBP^; hence P^P„^^JBP^. This,

however, is absurd, for a point separated from I^hy B andiji+i would have to belong to both classes. Our theoremresults from this contradiction.

We might treat the case where JJP Pji^ in exactly the

same way. Our net result is that if P be any point of the

line 2g, it is either a point of the system we have constructed,


or else we may find two such successive integers (calling ^ an

integer) n,n + l that F„ i^+jJPP^

.

Our next care shall be to find points of the line to which wemay properly assign fractional subscripts. Let Z^ be the linefrom P^ to the intersection of OP,, with VP^. Then I saythat F^ and 0^+j meet on Zj. This is certainly true whenk = 1. Let us assume it to be true in the case of l,i_i, sothat VII *^*i 0^ meet on Zj.i. Then Zj. is constructed withregard to Zj_j as was l^ with regard to l^, for we take a pointof l^_^, connect it with and find where that line meetsVI^. Jn like manner FJ^ meets 0^+^ on l^._i and 01^+ôn Ij, and so on; VI^ meets OP^+j, on Ij,, which was to beproved.As an application of this we observe that l„ meets VI^

on the line 0.^„, hence we easily see that ij, and P^ arehai-monically separated by ^ and ^„. Secondly, find thepoints into which the points Pj,i^,Pj are projected from onthe line FJ^. These points lie on the lines ^j.^, l,e^m> h-m'Find the intersections of the latter with VP^ and project backfrom on ô; we get the points i^+j_„, -^+fc_m, Pn-^-i-m-

A particular result of this will be that ^ Ih+n -^+2n -^ forma harmonic set.

Let us now draw a line from E to the intersection of VIând l„ , and let this meet ^ F in T^. Then if ij, Pj. , Pj , P„ be

nprojected from upon .^Fand then projected back from V^

n

upon Zp, we get points which we may call ^, i^, Pj, i^ , where

^ = ij. Connect P^ with the intersection of F^ and OP^ byn n

a line l^. We may use this line to find i^ as formerly we» »

used 2, to find ^. We shall thus find that Pg and i^„ are

harmonically separated by i^ and ^, or i^^ is identical» n

with ^, and similarly^ is identical with I^. SubdividingIT

still further we shall find that ^ is identical with i^ or^TO » TM

identical with .^. We have thus found & single definite

ti

point to correspond to each positive rational subscript.

Negative rational subscripts might be treated in the sameway, and eventually we shall find a single point whose sub-


script is any chosen rational number. We shall also find,

by reducing to a common denominator, that if

q>p>Q, P,Pq\^PpF^,

with a similar rule for negative nimibers.

It remains to take up the irrational case. Let P be anypoint of the positive separation class determined by ^ and P^

.

Then either it is a point with a rational subscript, according

to our scheme, or else, however great soever n may be, we

may find m so that P„p{p^T^, P^P^^^{PP^. We thusJ n n J

have a dichotomy of the positive rational number system

of such a nature that a number of the lower class will

correspond to a point separated from i^ by Jj and P while

one of the upper class will correspond to a point separated

from i^ by P and P^. There wiU be no largest number in

the lower class. We know, in fact, that wherever R maybe in the positive separation class of ijij^ we may find n'

80 great that î^ Jii^. We may express this by saying

that P„, approaches i^ as a limit as n' increases. Hence,as separation is invai-iant under projection, l^ approaches i^

as a limit and P^ approaches P, as a limit, or .^ j approaches

i^ as a limit. We can thus find n' so large that i^ ^ is alson n fv

a number of the first class, and surely — H—r > — • In the

same way we show that there can be no smallest numberin the upper class. Finally each number of the upper is

greater than each of the lower. Hence a perfect dichotomyis effected in the system of positive rationals defining a precise

irrational number, and this may be assigned as a subscript

to P. A similar proceeding will assign a definite subscript to

each point of the other negative separation class oil^P^.Conversely, suppose that we have given a positive irrational'

number. This wul be given by a dichotomy in the system of

positive rationals, and corresponding thereto we may establish

a classification among the points of the positive separationclass of ^.^ according to the requirement of Axiom VII.We shall, in fact, assign a point A of this separation class

to the lower sub-class u we may find such a number in thelower number class that the point with the corresponding


subscript is separated from J^ by i^ and A ; otherwise a pointshall be assigned to the upper sub-class. If thus A and Bbe any two points of the lower and upper sub-classes respec-

tively, we can find — in the lower number class so that

Po 4, Up^ whereas P, B \p^ P„ , and, hence, by 3, P^B \aP^ .

This shows that all of the requirements of Axiom YU are

fulfilled, we may assign as subscript to the resulting point

of division the iiTational in question. In the same way wemay assign a definite point to any negative irrational Theone to one correspondence between points of a line and the real

number system including co is thus complete.

D^nition. If A, B, C, D be four collinear points, whereofthe first three are necessarily distinct, the subscript whichshould be attached to B, when A, B, are made to play

respectively the rdles of ^,i2,ij in the preceding discussion,

shall be called a cross ratio of the four given points, andindicated by the symbol (AB, CD). Four points which are

distinct would thus seem to have twenty-four different cross

ratios, as a matter of fact they have but six.

We know that the harmonic relation is unaltered by anyfinite number of projections and intersections. We may there-

fore define the cross ratios of four concurrent coplanar lines,

or four coaxal planes, by the corresponding cross ratios of

the points where they meet any other line.

Theorem 27. Cross ratios are unaltered by any finite

number of projections and intersections.

D^niiion. The range of all collinear points, the pencil

of all concuiTsnt coplanai- lines, and the pencil of coaxal

planes shall be called fundamental one-dimensionalforms.

Dejmition. Two fundamental one-dimensional forms shall

be said to be projective if they may be put into such a one to one

correspondence that corresponding cross ratios are equal.

Theorem 28. If in two projective one-dimensional forms

three elements of one lie in the corresponding elements of

the other, then every element of the first lies in the corre-

sponding element of the second.

For -we may use these three elements in each case as oo, 0, 1,

and then, remembering the definition of cross ratio, make use

of the &ct that the construction of the harmonic conjugate

b2


of a point with regard to two others is unique. This theorem

is known as the fundamental one of projective geometry.*

Theorem, 39. If two fundamental one-dimensional forms be

connected by a finite number of projections and intersections

they are projective.

This comes immediately from 27.

Theorem, 30. If two fundamental one-dimensional forms be

projective, they may be connected by a finite number of

projections and intersections.

It is, in fact, easy to connect them with two other projective

forms whereof one contains three, and hence all corresponding

members of the other.

Let us now turn back for a moment to our cross ratio scale.

We have already seen that in the case of integers, and, hence,

by reducing to least common denominator, in the case of

all rational numbers k, I, m, n.

By letting k, I, m, n become irrational, one at a time, andapplying a limiting process, we see that this equation is

always true.

In like manner we see that I^, 11,11-, ^ form a harmonic

set, as do 4, Pq+k,^q+k> -^ • ^ general, therefore,

{P^P„P,Fy)={P^P„P,P^)

= V.

Putting n+ a = 3, nv + a = y,

(P^P„,P,P^) = rz^^.

We next I'emark that the cross ratio of four points is thatof their harmonic conjugates with regard to two fixed points.B«verting to our previous construction for ij we see that it is

ncollinear with V^ and Q^. VQgP_i are also on a line. If,

nthen, we compare the triads of points VP^Q^, Tî^Qj, since

lines connecting corresponding points are concurrent in I^^,

the intersections of corresponding lines are collinear. But

* For an interesting historioal note concerning this theorem, see Vablen,loc, cit., p. 161.


the line from to the intersection of P[iJ with F^ (or VQ,)

18, by construction, the line 0P„. Hence FPj, which is

identical with VQ^, meets F^ J^ on OP^^. Furthermore andn n

Qi are harmonically separated by the intersections of theirline with VP_j^ and T(^ ; i.e. by ^ and the intersection with

FQ,. Project these four upon Ig from the intersection of 0ând VPLi. We shall find ^ and ij are harmonic conjugates

swith regard to JJ and P_j. Let the reader show that this last

relation holds equally when ti is a rational fraction, and,hence, when it takes any real value.

The preceding considerations will enable us to find the«ross ratio of four points which do not include P^ in their

number. To bîn with

(P.Pfl, P,S8) = (P„Pj, P,P,)

3 7 «

= — X •

Let us project our four points from F upon l„, then backupon Ig from 0. This will add a to each subscript. Thenreplace y + a by y, &c.

Tkewem, 31. Four elements of a fundamental one-dimen-sional form determine six cross ratios which bear to oneanother the relations of the six numbers

^' y ^-^' uTK' IT' x^rThe proof is perfectly straightforwai'd, and is left to the

reader.

If three points be taken as fundamental upon a straight

line, any other point thereon may be located by a pair of

homogeneous coordinates whose ratio is a definite cross ratio

of the four points. We shall assign to the fundamental points

the coordinates (1,0), (0, 1), (1, 1). A cross ratio of four points

(x), (y), (z), {t) will then be

(2)


Any projective transformation of the line into itself, i.e. anypoint to point transformation which leaves cross ratios un-

altered, will thus take the form

Fx^ = «oo*o+*oi''a>I a- I ^ (3)

To demonstrate this we have merely to point out that surely

this transformation is a projective one, and that we may so

dispose of our arbitrary constants as to carry any three distinct

J)ointB into any other three, the maximum amount of freedom

or any projective transformation of a fundamental one-

dimensional form. Let the reader show that the necessary

and sufficient condition that there should be two real self-

corresponding points which separate each pair of corresponding

points isI

^ ,. 1^

Two projective sets on the same fundamental one-dimen-sional form whose elements con-espond interchangeably, are

said to foi-m an involution. By this is meant that eachelement of the foim has the same corresponding elementwhether it be assigned to the first or to the second set.

It will be found that the necessary and sufficient conditionfor an involution in the case of equation (3) will be

ttoi = a,p. (4)

When the determinant|a,--

|> 0, there will be no self-

corresponding points, and the involution is said to be dliptic.

Let the reader show that under these circumstances each pairof the involution separates each other pair.

Our next task shall be to set up a suitable coordinatesystem for the plane and for space. Let us take in the planefour points A, B, C, D, no three being collinear. We shall

assign to these respectively the coordinates (1, 0, 0), (0, 1, 0),

(0, 0, 1), (1, 1, 1). Let AD meet BG in A^, BD meet OAin Bi,aad CD meet AB in C^. The intersections of AB, A-^B^,of BG, B^G-i, and of GA, GÂ^, are, by 15, on a line d. Nowlet P be any other point in the plane

{ABAG, ADAF) = (PC^PG, PDPA) = (PGPG^, PAPD){BGBA, BDBP) = {PGPG^, PDPB)

{GAGB, GDGP) = {PG^PC, PAPB) ={PGPG^, PAPB)

From this it is clear that the product of the three is equalto unity, and we may represent them by three numbers of the


Qi SC Q*

type -i, -? ,

-fi. We may therefore take a;„ : a;, : as, as three

"^ Xq Xi X^ ' 012homogeneous coordinates for the point P. One coordinatewill vanish for a point lying on one of the lines AB, BG, CA.Let the reader convince himself that the usual cartesian

system is but a special case of this homogeneous coordinatesystem where two of the four given points are ideal, and

^ = a;,'^ = y.

The equations of the lines connecting two of the points

A, B, G are of the formXi = 0.

Those which connect each of these with the point D are

similaxlyx,-a.,.=0.

If [y) and (z) be two points, not collinear with A, B,ot G,

while P is a variable point with coordinates X(i/) + ^(2), the

lines connecting it with A and B will meet BG and {GA)respectively in the points

(0, Xy^ + iiZi, Ky^+iiz^) {Xy^ + ixz^, 0, ky^+ixz^).

It is easy to see that the expressions for corresponding cross

ratios in these two ranges are identical, hence tiie ranges are

projective. The pencils which they determine at A and Bare therefore projective, and have the line AB self-correspond-

ing, for this will correspond to the parameter value

X:m = S2= -2/2-

But it will follow immediately from 28, that if two pencils

be coplanor and projective, with a self-corresponding line,

the locus of the intersection of their corresponding membersis also a line. Hence the locus of the point P with the

coordinates A.(3/) + m(2) is the line connecting (y) and (z).

Conversely, it is evident that every point of the line from

(y) to (z) will have coordinates linearly dependent on those

of (y) and (z). If, then, we put

«i = ^2/» + <*«».

and eliminate X : /x, we have as equation of the line

I

xyzI

= (ux) = 0.

Conversely, it is evident that such an equation will always

represent a line, except, of course, in the trivial case where

the u's are all zero. Let the reader show that the coefficients


Ui have a geometrical interpretation dual to that of the

coordinates a-^ ; for this purpose the line which we have above

called d will be found useful.

Our system of homogeneous coordinates may be extended

with great ease to space. Suppose that we have given five

points A, B, G, D,0 no four being coplanar. Let P be anyother point in space. We may write

{ABCABD, ABOABP) = ^ , {ACDACB, AGOAGP) = ^ ,

x^ a;,

{ABB ADC, ADOADP)^^.Xy

We shall then be able to write also

{ODA GDB, GDO GDP) = ^ , (DBADBG, DBO DBP) = ^ .

Xq ajg

(BCDBCA, BCOBGP)=^.

In other words, we may give to a point four homogeneouscoordinates x^-.x^-.x^-.x^. Two points coUinear with A, B,

G, or D will differ (or may be made to differ) in one coordinateonly. An equation of the first degree in three coordinates

will represent a plane through one of these four points.

Every line will be the intersection of two such planes, andwill be represented by the combination of two linear equations

one of which lacks Xf while the other lacks X;. The coor-

dinates of all points of a line may therefore be expressed as

a linear combination of the coordinates of any two pointsthereof. A plane may be represented as the assemblage of

all points whose coordinates are linearly dependent on thoseof three non-collinear points. Eliminating the variable para-meters from the four equations for the coordinates of a pointin a plane, we see that a plane may also be given by anequation of the type / x « ,„>^ •'*^

(ux) = 0. (5)

Conversely, the assemblage of all points whose coordinatessatisfy an equation such as (5) will be of such a nature thatit will contain all points of a line whereof it contains twodistinct points, yet will meet a chosen line, not in it, butonce. Let the reader show that such an assemblage mustbe a plane. The homogeneous parameters (u) which, naturally,may not all vanish together, may be caJled the coordinates


of the plane. They will have a significance dual to thatof the coordinates of a point.*

If we have four collinear points

(2/). (2), Hy)+f^(^), y{y)+h'(^),

one cross ratio will be Ajm'

The proof will consist in finding the points where fourcoaxal planes through these four points meet the line

ojg = aJs = 0,

and then applying (2).

Suppose that we have a transformation of the type

0..S

pXi'=âijXj. (6)

j

This shall be called a colli/neation. We shall restrict

ourselves to those collineations for which

i

ayi

gt 0.

The transformation is, clearly, one to one, with no ex-ceptional points. It will carry a plane into a plane, a line

into a line, a complete quadrilateral into a complete quadri-

lateral, and a harmonic set into a harmonic set. It will

therefore leave cross ratios invariant. Moreover, every point

to point and plane to plane transformation will be acollineation. For every such transformation will enjoy all

of the properties which we have mentioned with regard to

a collineation, and will, therefore, be completely detei-mined

when once we know the fate of five points, no four of whichare coplanar. But we easily see that we may dispose of the

arbitrary constants in (6), to carry any such five points into

any other five.

It is worth while to pause for a moment at this point in

order to see what geometi-ical meaning may be attached to

coordinate sets which have imaginary values. This question

* The treatment of cross ratios in the present chapter is based on that of

Pasch, loe. cit. The development of the coordinate system is also taken fromthe same source, though it has been possible to introduce notable simplifica-

tion, especially in three dimensions. This method of procedure seemed to

the anihor more direct and natural than the more modem method of' Streckenrcchnung ' of Hilbert or Vahlen, loc. cit.


has already been discussed in Chapter VII. Every set of

complex coordinates (y)+i(z)

may be taken to define the elliptic involution

(x) = \{y) + ixiz), x = k'(y) + ^'(z), \\' + (i^'=0. (7)

To verify this statement we have merely to notice that an

involution will, by definition, be canied into an involution

by any number of projections and intersections, and that

equations such as (7) will go into other such equations. But

in the case of the line jp _ ^ _ q

these equations will give an involution, for the relation

between (x) and (a;') may readily be reduced to the type of (3)

and (4). Did we seek the analytic expression for the coor-

dinates of a self-corresponding point in (7) we should get

the values{y) + i(z).

Conversely, it is easy to show that any elliptic involution

may be reduced to the type of (7). There is, therefore, a oneto one correspondence between the assemblage of all elliptic

point involutions, and all sets of pairs of conjugate imaginarycoordinate values.

The correspondence between coordinate sets and elliptic

involutions may be made more precise in the following fashion.

Two triads of collinear points ABC, A'B'C shall be said to

have the same sense when the projective transformation whichcarries the one set, taken in order, into the other, has a positive

determinant ; when the determinant is negative they shall besaid to have opposite seTises. In this latter case alone, as wehave already seen, will there be two real self-corresponding

points which separate each distinct pair of correspondingpoints. Two triads which have like or opposite senses to

a third, have like senses to one another, for the determinantof the product of two projective transformations of the line

into itself is the product of the determinants. We shall alsofind that the triads ABC, BOA, CAB have like senses, whileeach has the sense opposite to that of either of the triads

ACB, CBA, BAC. We may thus say that three points givenin order will determine a sense of description for the wholerange of points on the line, in that the cyclic order of anyother three points which are to have the same sense as thefirst three is completely determined. It is immediatelyevident that any triad of points and their mates in anelliptic involution have the same sense. We may therefore


attach to such an elliptic involution either the one or theother sense of description for the whole range of points.

D^nition. An elliptic involution of points to which is

attached a particular sense of description of the line on whichthey are situated shall be defined as an imaginary point.The same involution considered in connexion with the othersense shall be called the corijugate ivnaginary poiint.

Starting with this, we may define an imaginary plane asan elliptic involution in an axial pencil, in connexion witha sense of description for the pencil; when the other senseis taken in connexion with this involution we shall say thatwe have the conjugate imaginary plane. An imaginary pointshall be said to be in an imaginary plane if the pairs of theinvolution which determine the point lie in pairs of planesof the involution determining the plane, and if the sense of

description of the line associated with the point engendersamong the planes the same sense as is associated with theimaginary plane. Analytically let us assume that besides

the involution of points given by (7) we have the foUowinginvolution of planes.

(it) = l{v) + m{w), {u') = l'{v)+m' {w), IV +mm'= 0,

{vy) = (loz) = 0. (8)

The plane (u) wiU contain the point I (vz) (y) —m (wy) (z)

while its mate in the involution contains the point

m(vz)(y) + l(wy)(z).

These points will be mates in the point involution, if

[{vz) + (wy)] l(vz)- (ivy)] = 0,

and these equations tell us that the imaginary plane (v) + i (w)

will contain either the point (y) + i{z), or the point {y)—i{z).

An imaginary line may be defined as the assemblage of all

points common to two imaginary planes. Imaginary points,

lines, and planes obey the same laws of connexion as doreal ones. A geometric proof may be found based upon the

definitions given, but it is immediately evident analytically.*

Theorem 32. If a fundamental one-dimensional form beprojectively transformed into itself there will be two distinct

or coincident self-corresponding elements.

We have merely to put (px) for («') in (3), and solve the

* See von Staudt, loc. cit., and Luroth, loc. cit. It is to be noted that inthese works the idea of sense of description is taken intuitively, and not givenby precise definitions.

268 THE PROJECTIVE BASIS OF oh.»

quadratic equation in p obtained by equating to zero the deter-

minant of the two linear homogeneous equations in a;,, XyThe assemblage of all points whose coordinates satisfy an

equation of the type

shall be called a quadric. We should find no difficulty in

proving all of the well-known theorems of a descriptive sort

connected with quadrics in terms of our present coordinates.

We have now, at length, reached the point where we mayprofitably introduce metrical concepts. Let us recall that the

group of congruent transformations which we considered in

Chapter 11, and, more fully, in Chapter VIII, is a group of

collineations which leaves invariant either a quadi-ic or a

conic, and depends upon six parameters. We also saw in

Chapter II, that the congruent group may be characterized

as follows (cf. p. 38):

—

(a) Any real point of a certain domain may be carried into

any other such point.

(b) Any chosen real point may be left invariant, and anychosen real line through it carried into any other such line.

(c) Any real point and line through it may be left invariant,

and any real plane through this line may be carried into anyother such plaae.

(d) If a real point, a line through it, and a plane throughthe line be invariant, no further infinitesimal congruenttransformations are possible.

It shall be our present task to show that these assumptions,or rather the last three, joined to the ones already made in

the present chapter, will serve to define hyperbolic elliptic

and euclidean geometry.It is assumed that there exists an assemblage of transforma-

tions, called congruent transformations, obeying the followinglaws:

—

Axiom Xn. The assemblage of all congruent transforma-tions is a group of collineations, inoluding the inverse ofeach member.*

* It is highly remarkable that this axiom is superfluous. Cf. Lie-Engel,Uteorie der Trantformationsgruppen, Leipzig, 1888-93, vol. iii, Ch. XXII, $ 98.The assumption that our congruent transformations are collineations, does,however, save an incredible amount of labour, and, for that reason, is in-cluded here.


Axiom XIII. The group of congruent transformations maybe expressed by means of analytic relations among theparameters of the general collineation group.

Befinition. The assemblage of all real points whose co-ordinates satisfy three inequalities of the type

f< < J < Z,, i = 1, 2, 3,

shall be called a restricted region,.

Axiom XIV. A congruent transformation may be foundleaving invariant any point of a restricted region, andtransforming any real line through that point into any othersuch line.

Axiom XV. A congruent transformation may be foundleaving invariant any point of a restricted region, and anyreal line through that point; yet carrying any real planethrough that line into any other such plane.

Axiom XVI. There exists no continuous assemblage ofcongruent transformations which leave invariant a point of

a restricted region, a real line through that point, and a real

plane through that line.

Theorem 33. The congruent group is transitive for a suffi-

ciently small restricted region.

This comes at once by red/uctio ad absurdum. For the

tangents to all possible paths which a chosen point mightfollow would, if 33 were untrue, geneitite a surface or set

of surfaces, or a line or set of lines, and this assemblage of

surfaces or lines would be carried into itself by every con-

gruent transformation which left this point invariant. Thetangent planes to the surfaces, or the lines in question, could

not, then, be freely interchanged with other planes or lines

through the point.

Theorem 34. The congruent group depends on six essential

parameters.

The number of parameters is certainly finite since the

congruent group arises from analytic relations among the

fifteen essential parameters of the general collineation group.

The transference from a point to a point imposes three

restrictions, necessarily distinct, as three independent para-

meters are needed to determine a point. A fixed point being

chosen, two more independent restrictions are imposed by


determining the fate of any chosen real line through it.

When a point and line through it are chosen, one more

restiiction is imposed hy determining what shall become of

any assigned plane through the line. When, however, a real

plane, a real line therein, and a real point in the line are

fixed, there can be no independent parameter remaining, as no

further infinitesimal transformations are possible.

Let us now look more closely at the one-parameter family

of projective transformations of the axial pencil through

a fixed line of the chosen restricted region.* Let us deter-

mine any plane through this line by two homogeneousparameters AîXj, and take an infinitesimal transformation

of the group , i . ,\ .

The product of two such infinitesimal transformations will

belong to our group, hence also, as none but analytic functions

are involved, the limit of the product of an infinite numberof such transformations as dt approaches zero ; that is to say,

the transformation obtained by integrating this equationbelongs to the group. Now this integral will involve onearbitrary constant, which may be used to make the transfor-

mation transitive, and for all transfoimations obtained bythis integration, that pair of planes will be invariant whichwas invariant for the infinitesimal transformation. Ourone-parameter group has thus a transitive one-parameter sub-group with a single pair of planes invariant. These planes

ai-e surely conjugate imaginary, for otherwise there wouldbe infinitesimal congruent transformations which left a point,

line, and real plane invariant; contrary to our last axiom.The question of whether our whole one-parameter group is

generated by this integration or not, need not detain us here.

What is essential is that this pair of planes will be invariantfor the whole group. For suppose that S^ indicate a generictransformation of the sub-group which leaves invariant thetwo planes a, a', and the transformation T carries the twoplanes a, a' into two planes fi, jS'. Then all transformationsof the type TS-T~^

wUl belong to our group, and leave the planes y3,p' invariant,

and combining these with the transformations 8^ we havea two-parameter sub-group of our one-parameter group; anabsurd result.

* Cf. Lie-Scheffers, Vorlesungen iiber coidinuierliche Grupptn, Leipzig, 1893,p. 126.


Let us next consider the three-parameter congraent groupcomposed of all transformations which have a fixed point.If a real line I be carried into a real line V, then the twoplanes which were invariant with I will go into those whichare invariant with V. To prove this we have but to repeatthe reasoning which lately showed that the two planes whichwere invariant for a sub-group, are invariant for the total

one-parameter group. The envelope of all these invariantplanes which pass through a point will thus depend uponone parameter, for if it depended on two it would includereal planes, and this is not the case. It is well known thatthis system of planes must envelope lines or a quadric cone.*The first case is surely excluded for such lines would haveto appear in conjugate imaginary pairs, giving rise to in-variant real planes through this point, and there are no suchin the three-parameter group. The envelope is therefore

a cone with no real tangent planes. Each pair of conjugateimaginary tangent planes must touch it along two conjugateimaginary lines ; the plane connecting these is real, andinvariant for the one-parameter congruent group associated

with the line of intersection of the two imaginary planes.

Let us fix our attention upon one such one-parameter groupand choose our coordinate system in such a way that the

non-homogeneous coordinates u, v, 1 of our thi-ee fixed planes

are proportional respectively to

(0, 0. 1), (1, i, 0), (1, -i, 0).

The general linear transformation keeping these three

invariant is

vf—rcaaBu—rsaxiOv, i/= r sin tfu + r cos flw.

Here r must be a constant, as otherwise we should havecongruent transformations of the type

«'= rw, v'=rv,

which kept a point, a line, and all planes through that line

invariant, yet depended on an arbitrary parameter. In order

to see what sort of cones are carried into themselves by this

group, the, cone we ai"e seeking for being necessarily of the

number, let us take an infinitesimal transformation

A% = —vd6, Av = vdd.

Integrating ^2^^ _ cr

The cone we seek is therefore a quadric cone.

* Cf. Lie-Scheffers, loc; cit., p. 289.


We see by a repetition of the sort of reasoning given abovethat if we take a congruent transformation that carries

a point P into a point P', it will carry the invariant quadric

cone whose vertex is P into that whose vertex is P\ Theenvelope of these quadric cones is, thus, invariant under the

whole congruent group. The envelope of these cones mustbe a quadric or conic. This theorem is simpler when put

into the dual form, i.e. a surface which meets every plane

in a conic is a quadric or quadric cone. For it has just the

same points in every plane as the quadric or cone throughtwo of its conies and one other of its points. In our present

case our quadric must have a real equation, since it touches

the conjugate to each imaginary plane tangent thereto. Thereare, hence, three possibilities

:

(a) The quadric is real, but the restricted region in question

is within it.

(b) The quadric is imaginary.

(c) The quadric is an imaginary conic in a real plane.

Theorem, 35. The congruent group is a six-parameter colli-

neation group which leaves invariant a quadric or a conic.

It remains for us to find the expression for distance. Wemake the following assumptions.

Axiom XVII. The distance of two points of a restnctedregion is a real value of an analytic function of their

coordinates.

Axiom XVIII. If ABC be three collinear real points, andif £ be separated by A and G team a point of their line notbelonging to this restricted region ; then the distance from.^ to O is the sum of the distance firom A to B and thedistance from B to G.

Let the reader show that this definition is legitimate as all

points separated from A hy B and C, or from Chy A and Bwill belong to the restricted region.

Let us first take cases (a) and (b) together. The distancemust be a continuous function of each cross ratio determined bythe two points and the intersections of their line with thequadric. If we call a distance d, and the corresponding crossratio of this type c, we must have

c=f(d).

Moreover, from equation (1) and Axiom XIII,

f{d)xf{d')=f{d + d').


Now this fanctional equation is well known, and the onlycontinuous solution is*

d

c = e *.

d 1

If, in particular, the two points be P^P^ while their linemeets the quadric in Q^Q^, we shall have for our distance,equation (5) of Chapter VII

From this we may easily work back to the familiar ex-pressions for the cosine of the ^th pai-t of the distance.

The case of an invariant conic is handled somewhatdifferently. Let the equations of the invariant conic be

Xg = 0, x-^ + x^ + x^ = 0.

These are unaltered by a seven-parameter group

•"3 ^^ **30*0'^''^S1^1 '"*32'''2"'''''33*''3'

where ||ajia22<^|| is the matrix of a ternary orthogonalsubstitution. For our congruent group we must have thesix-parameter sub-group where the determinant of this ortho-

gonal substitution has the value a^, for then only will there

be no further infinitesimal transformations possible whena point, a line through itj and a plane through the line aiê

fixed. We shaU find that, under the present circumstances

the expression

D = I

J'h - yi)\ p -¥)%:p1^'I V Va;o Vo^ Va!o y^' ê^ Vo'

is an absolute invariant. If the distance of two points (x), (yy

be d, we shall have d = f(D).

This function is continuous and real, and satisfies the

functional equation

f(D)+f{iy)=fiD + D').

* Cf. e.g. Tannery, Thiorit desfonctions cCune variaWe, second edition, Paris,.

1904, p. 275.

OOOLIDOB S

274 PROJECTIVE BASIS ch. xviii

The solution of this equation is easily thrown back uponthe preceding one. Let us put

f{x) = log4>(x),

4,(x)<l>(y) = il>{x + y),

<l>(x) = c'»'.

We thus get finally

I V ^0 yJ ^0 yJ ^0 y^'

Theorem 36. Axioms I-XVIII are compatible with the

hyperbolic, elliptic, or euclidean hypotheses, and with these

only.

CHAPTER XIX

THE DIFFERENTIAL BASIS FOU EUCLIDEANAND NON-EUCLIDEAN GEOMETRY

We saw in Chapter XV, Theorem 17, that the Gaussiancurvature of a surface is equal to the sum of the total relativecurvature, and the measure of curvature of space. A non-euclidean plane is thus a surface of Gaussian curvature equal

to p - This fact was also brought out in Chapter V, Theorem 3,

and we there promised to return in the present chapter toa more extensive examination of this aspect of our non-euclidean geometry.

In Chapter n, Theorem 30, we saw that the sum of thedistances from a point to any other two, not; collinear withit, when such a sum exists, is greater than the distance ofthese latter. We thus come naturally to look upon a straight

line as a geodesic, or curve of minimum length between twopoints. A plane may be generated by a pencil of geodesies

through a point ; the geometrical simplicity of the plane maybe said to arise from the fact that it is capable of x' suchgenerations. The task which we now undertake is as

follows:—to determine the nature of a three-dimensional

point-manifold which possesses the property that eveiy sur-

face generated by a pencil of geodesies has constant Gaussiancurvature. We must begin, as in previous chapters, witha sufficient set of axioms.*

Definition. Any set of objects which may be put into one

to one correspondence with sets of real values of three inde-

pendent coordinates Zi,z^, z^ shall be called poinis.

DefiTvition. An assemblage of points shall be said to forma restricted region, when their coordinates are limited merely

by inequalities of the type

Ci<Zi<Zi, i=l,2,3.

* The fiist writer to approach the snbject from this point of view wasRiemann, loc. cit. The best presentation of the problem in its general form,

and in a space of n-dimensions, will be found in Schur, ' Ueber den Zusam-menhang der BSume constanten Biemannschen Kriimmungsmasses mit denprojectiven Bamnen,' Matkematiache Annalen, toI. 27, 1886.

S2

276 THE DIFFERENTIAL BASIS FOR EUCLIDEAN ch.

Axiom I. There exists a restricted region.

Axiom II. There exist nine functions a^ , i, j = 1, 2, 3

of :?!, ^2, Sj real and analytic throughont the restricted region,

and possessing the following properties

ay = aji, ! Uij ! ^ 0.

1,2,3

'j

is a positive definite form for all real values of dz^, dz^, dz^

and all values of 2:,, Zj' ^3 corresponding to points of the given

restricted region.

Limitation. We shall restrict ourselves to such a portion

of the original restricted region that for no point thereof shall

the discriminant of our quadratic form be zero. This amountsto confining ourselves to the original region, or to a smaller

restricted region within the original one.

Definition. The expression

1, -i, 8

d8=: +^ '^ttijdZidZj

shall be called the distance element.

Definition. The assemblage of all points whose coordinates

are analytic functions of a single parameter shall be called ananalytic curve, or, more simply, a curve. As we have defined

only those points whose coordinates are real, it is evident that

the functions involved in the definition of a curve must bereal also. The definite integral of the distance element

between two chosen points along a curve shall be called the

length of the corresponding portion or arc of the curve. If

the curve pass many times through the chosen points, the

expression length must be applied to that portion along whichthe integration was performed.

Definition. An arc of a curve between two fixed points

which possesses the property that the first variation of its

length is zero, shall be called geodesic arc. The curve whereonthis arc lies shall be called a geodesic connecting the twopoints.

XIX AND NON-EUCLIDEAN GEOMETRY 277

Let us begin by setting up the differential equations for

a geodesic. Let us write

It is dear that s is an analytic function of t with nosingularitieB in our region, hence t is an analytic function of 6.

We may, then, by taking our restricted region sufficiently

small, express a,-.- as functions of 8, and write

^^•* dzidz-

dz-'"'

Replacing -r-* temporarily by 2/, we have

We have now a simple problem in the calculus of

variations.

-,1,2,3 1,2.3 J_..2 6fi =

J2 J,{jfziz/^'îc + 2aijz/bz/)ds.

J 1,2,3 l,2,SJ/„. _/\ 1,2,3

hence, since 6r • vanishes at the extremities of the interval

. ,1, 2, 3 \i, 2, 3 -. J I

-Xlj \_li J i J

the increments 82 • are arbitrary, hence the coefficients of each

must vanish, or

d v'„ dzi_l''4^'iaij,iziizu^2)^ Z "'ij dB~%Z 2,2. Js as ^ '

These three equations are of the second ordei-. There will

exist a single set of solutions corresponding to a single set

of initial values for (2) and (2').* Let these be (2") and (Q

* Cf. e.g. Jordan, Cours SAnvisx, Paris, 1893-6, vol. iii, p. 88.

278 THE DIFFERENTIAL BASIS FOR EUCLIDEAN CH.

respectively. Any point of such a geodesic -will be determined

by Ci Co (3 ^^^ '"' *^® length of the arc connecting it yiith (z").

We have thus12 3

D (z z z )

Now the expression n , \^ '..

,

has the value unity when

r = 0. We may therefore revert our series, and write

12 3

»-Ci = 2~V+ 2^jJt (^j-y) («)fc-V) + ••• (4)

We shall take our restricted region so small that (4) shall

be uniformly convergent therein, for all values for (z) and (nfi)

in the region. Hence two points of the region may be con-

nected by a single geodesic €u:c lying entirely therein.*

Tkeorevn 1. Two points of a restricted region whose coor-

dinates differ by a sufiSciently small amount may be connected

by a single geodesic arc lying wholly in a sufficiently small

restricted region which indudes the two points.

We shall from now on, suppose that we have limited

ourselves to such a small restricted region that any twopoints may be so connected by a single geodesic arc.

Defiivition. A real analytic transformation of a restricted

region which leaves the distance element absolutely invariant

shall be called a congruent transformation.

Definition. Given a geodesic through a point (z**). Thethree expressions

shall be called the direction coamea of the geodesic at thatpoint. Notice that

1,2,3.

1,2.S 1,2,3_ _

/l, 2,8 \2

1.2.S

= 2 ifîiîi-<î^) (CiCj-CjQ*.

* Cf. Darboox, loc. eit, vol. ii, p. 408.


This is a positiTe definite fonn, for the coefficients are theminois of a positive definite form. Hence

1,2.3

,''

This expression shall be defined as the cosine of the angleformed by the two geodesies. When it vanishes, the geodesiesshall be said to be mutually perpendicviar or to cut at rightangles.

Theorem, 2. The angle of two intersecting geodesies is anabsolute invariant for all congruent transformations.

This comes at once from the fact that

1,2,8

â^jdzilzj

is obviously an absolute invariant for all congruent trans-formations.

DefinMion. A set ofgeodesies through a chosen point whosedirection cosines there, are linearly dependent upon thoseof two of their number, shall be said to form a pencil. Thesurface which they trace shall be called a geodesic aiwrface.

We shall later show that the choice of the name geodesicsurface is entirely justified, for each surface of this sortmay be generated in oo* ways by means of pencils ofgeodesies.

Axiom IIL There exists a oongrnent transformationwhich carries two sufficiently small arcs of two intersecting

geodesicB whose lengths are measured firom the commonpoint, into two arcs of equal length on any two inter-

secting geodesies whose angle is equal to the angle of theoriginal two.*

It is clear that a congruent transformation will carry anarc whose variation is zero into another such, hence a geodesic

* Our Axioms I-IU, are, irith slight verbal alterations, those used byWoods, loc. cit. His article, though yitiated by a certain haziness of defini-

tion, leaves nothing to be desired from the point of view of simplicity. Inthe present chapter we shall use a different coordinate system from his, in

order to avoid too close plagiarism. It is also noteworthy that he uses k

where we conformably to our previous practice use -•


into a geodesic. It will also transform a geodesic surface

into a geodesic surface, for it is immediately evident that

we might have defined a geodesic surface as generated by

those geodesies through a point which are perpendicular to

a chosen geodesic through that point.

It is now necessary to choose a particular coordinate system,

and we shall make use of one which will turn out to be

identical with the polar coordinate system of elementary

geometry. Let us choose a fixed point (z"), and a fixed

geodesic through it with direction cosines (C). Finally, wechoose a geodesic surface determined by our given geodesic,

and another through (2"). Let<t>be the angle which a geodesic

through (2") makes with the geodesic (C), while d is the angle

which a geodesic perpendicular to the last chosen geodesic

and to (f) makes with a geodesic perpendicular to the given

geodesic surface, i.e. perpendicular to the geodesies of the

generating pencil. Let r be the length of the geodesic arc of

(Q from (z") to a chosen point. We may take <^, 6, r as coor-

dinates of this point. The square of the distance element

will take the form

d^ = dr^ + Ede^ + 2Fded<f> + Gdcjyl (5)

We see, in fact, that there will be no term in drd<l> or drdO.For if we take 6 = const, we have a geodesic surface, andthe geodesic lines of space radiating from (2°) and lying in

this surface will be geodesies of the surface. The curves

r = const, will be orthogonal to these radiating geodesies.*

The surfaces ^ = const, are not geodesic surJQEMses, but the

curves 6 = const, and r = const, form an orthogonal system for

the same reason as before. The coefficients E, F, G are indepen-dent of 0, for, by Axiom III, we may tiunsform congruentlyfrom one surface = const, into another such. The coefficient

G is independent of tf> also, for in any surface 6 = const,

we may transform congruently from any two geodesies

through (z") into any other two making the same angle.

We may, in fact, write

E=G(T)E'(<p), F=G(T)F(i,),

for the square of any distance element can be put into the

formd^ = dr^-\-Gd^,

where </>] is a function of and 0.

* Bianchi, DiffarmtialgeomelTie, cit., p. 160.


Let us at this point rewrite our differential equations (2)in terms of our present coordinates

dalda} 2\j>r\ds) '^ lr\ds)\ds) '^ lr\ds)

y

dsV^ d^^"^ ds\ - 2ll^\dii)^'*

M' \d8)\d^)\'

(6)

Consider the geodesic surface <^ = - , which may, indeed,

be taken to stand for any geodesic surface. Here we must

where c is constant. The differential equations for a geodesiccurve on this surface will be *

dsVd8\~ 2llr\ds) ydJ„d6-\daX

These are exactly equivalent to the combination of (6) and</j = const. Lastly, if we remember that two near points of

a surface can be connected by a single geodesic arc lying

therein.

Theor&m 2. The geodesic connecting two near points of

a geodesic surface lies wholly in that surface, and is identical

with the geodesic of the surface which connects those twopoints.

Theorem, 3. There is a group of oc^ congruent transforma-

tions which carry a geodesic surface transitively into itself.

TheoreTti 4. All geodesic surfaces have the same constant

Gaussian cuiTature.

These theorems enable us to solve completely our differential

equations (6). The Gaussian curvature of each geodesic

surface is an invariant of space which we may call its

measure of curvature. We shall denote this constant by t^ ,

and distinguish with care the two following cases

p#0. ^, = 0.

* Bianchi, ibid., p. ISS.

282 THE DIFFERENTIAL BASIS FOR EUCUDEAN ch;

The determination of our coefficients E, F, G iB now aneasy task. The square of the distance element for a geodesic

surface 6 = const. , will be ds'^=:dr^ + G(r) d^\

Writing that this shall have Gaussian curvature ^ > we get

VG = Asihy +B cos i •

A; k

The determination of the constants A, B requires a little

care. It is clear to begin with that when

r = 0, G=0.Hence 5 = 0,

Again i.z.s %, ->„ 1,2,8 -.^ ^^

But, from (1)

1,2,3

1 = 2'^vCO = 2 «.(C+ ^*^0)(C+ ^c?.^).

•.J

1,2,3

d0 /V-' d(^

cos d^ = 2 Oy Ci (Cj- + jj <^*)

.

*=°'T = ^-2 2««a^4''*.

giving eventually

(^) =1; A = k.

Hence, by the equations preceding (6)

d^ = dr'' + k^ sin"^{I^dd^+ 2F'd<t>d(t> + d^,^.

XIX AND NON-EUCLIDEAN GEOMETRY 28S

We proceed to calculate P. The differential equations fora geodesic curve of the surface d = const;, will be

rfsVds/ 2drVds/ '

aC^S)'"-These must be equivalent to those obtained from (6), when= const., i.e. we must have

F'= const.,

and as F' is not a function of 5 it is a constant everywhere.Now when ^ = 0, there is no dO term in ds", so that E=0;

Ebut » which is the cosine of the angle which curves

= const, and ^ = const., make on the surface r = const.,

is surely less than unity. Hence

J"=0.

Lastly, we must find E'. The surfaces r = const, haveconstant Gaussian curvature, for each is capable of co^ con-gruent transformations into itsel£ Hence

ds" = B sin2^{E'd6^ + d<f>^,

1 dVF—;= = const.,-/BT df^

VE = Asinlip + B cos l(f>.

As we saw a moment ago B = 0, for E vanishes with ^.

On the other hand, when

But also A sin Iv = 0.

Hence Hs an old integer, and

ds« = dr'+P sin« ^ [sin" <l>d6^+ dify^. (7>


This is our ultimate form for the square of the distance

element. Let the reader show that under the second case

p = 0, we have

ds^ = dr^ + 1^ [sin*^ de^ + di>^\ (7')

It is now time to return to coordinates of a more familiar

sort. Let us write

7 ^'a;,= K cos r )

ajj = 4 sin t cosfl cos 0,

TXj= 4 sin r sinfl cos<^,

(8)

Xgzx^sin^^sin^,

{xx) = k^,

(dxdx) = ds^.

To find the differential equation of a geodesic, we havea problem in relative minima

i(^) = ^^^^' i = 0,1.2,3.

To determine A

{xx)=k\ {xdx)=-^dii^,

{xd^x) \-ds^= d(- ^ds«) = 0.

But from our equations

{xd^x)-%Kk^db^,

We thus get for the final form for our differential equation

d^Xs Xi

-d^^k^ = ''- (9)

Let the reader show that in the other case we have

d^x _ <Py _ d-z _ ,

ds^ ~W~ d^~^- <^

)

XIX AND NON-EUCLIDEAN GEOMETKY 285

Integrating s . s

B = (xx) = (yy) = (zz),

(yz) = 0.

We have then for the length of the geodesic arc from (y)to (a:) ^P cos ^ = {ay),

or, if we replace our coordinates by homogeneous ones pro-portional to them /J /™a

cosg= ^_^\ • (10)« V{xx) V(yy)

Let the reader show that when p = 0,

Theorem 5. Axioms I, II, HI ai-e compatible with theeuclidean hyperbolic and elliptic hypotheses, and with thesealone.

Our task is now completed. At bottom, the essential

feature of a geometrical system where the elements are pointsis the expression for distance, for the projective theory is

the same for a limited domain in all restricted regions. Wehave established our distance formulae three several times,

each time approaching the subject from a new point of view.In Chapters I-IV we took as fundamental the concepts point,

distance, and sum of distances. We reached our analytic

formulae by proceeding from elementary geometry to trigono-

metry, and then introducing a simple coordinate system, such

as we do when we fii'st take up the study of elementaryanalytic geometry. The Chapters VI-XVII were devoted to

erecting a superstructure upon the foundation which we hadestablished. In Chapter XVIII we took a fresh start, laid

down point line and separation as fundamental, constructed

the common projective geometry for all of our systems (except

the ^herical, which would involve slight modifications), andestabUshed the system of projective coordinates. We then

introduced certain collineations called conynient transforma-

tions, and worked around to our previous distance formulae

through group-theory. In the present chapter we took as

fundamental the concepts point and correspondence of point

and coordinate set. The essentials in our development were

the distance element, the geodesic curve, and the space con-

286 DIFFERENTIAL BASIS CH. xix

stant, or measure of curvature. We reached our familiar

formulae by means of surface theory, integration, and the

calculus of variations.

Which of the three methods of approach is the best 1 Tothis question no definite answer may be given, for that methodwhich is best for one purpose is not, necessarily, best for

another. The first method depended upon the simplest andmost natural fundamental conceptions, and presupposed aminimum of mathematical knowledge. It also correspondedmost closely to the line of historical development. On the

other hand it is the longest, even after cutting out a numberof theorems, interesting in themselves, but not essential as

steps towards the ultimate goal. The second method possessed

the advantage of beginning with the assumptions wmch serve

as a btisis for the import^t subject of projective geometry;metrical ideas were grafted upon this stem as a naturaldevelopment. Moreover, the fundamental importance of thesix-p&rameter coUineation group which keeps a conic orquadric invariant was brought into the clearest light. Onthe other hand, we were obliged to develop a coordinatesystem, which to some readers might seem a trifle unnaturalor forced, and exposed ourselves to being put down amongthose whom the late Professor Tait has stigmatized as ' Thatsection of mathematicians for whom transversals and an-harmonic pencils have a, to us, incomprehensible chai-m'.* Ourthird and last method is, beyond a peradventure, the quickestand most direct ; and has the advantage of bringing out thefull significance of the space constant. It may, however,be urged with some justice, that too high a price has beenpaid for this directness, by assuming at we outset that spaceis something whose elements depend in a definite manner onthi-ee independent parameters. The modem tendency is to

take a more abstract view, to look upon space, in the last

analysis, as a set of objects which can be arranged in multipleseries.! The battle is more than half over when the coor-

dinate system has been set up.

No, there is no answer to the question which method ofapproach is the best. The determining choice among thethree, will, in the end, be a matter of personal aesthetic

preference. And this is welL Let us not forget that, inlarge measure, we study pure mathematics to satisfy anaesthetic need. We are fortunate when, as in the present case,

we are free at the outset to choose our line of approach.

* Tait, An Elementary Treatise an Quatemimu, third edition, Cambridge, 1890,p. 309.

t Cf. Busaell, loc. cit., p. 372.

INDEX

AbBolute, 88, 94, 95, 97, 98, 99, 101,102, 103, 106, 107, 110, 111, 113,116, 117, 118, 119, 124, 127, 129,132, 134, 138, 142, 143, 146, 152,154, 155, 157, 161, 162, 187, 205,226, 231, 232, 233, 284, 244, 246.

Actual elements, 85.Amaldi, 177.

Amplitude of tetrahedron, 179, 180,181.

Amplitude of triangle, 170, 171,172, 173.

Angle, interior and exterior, 30, 87,88,279.— null, 30.

— right, 32.— straight, 31.— re-entrant, 31.— dihedral, 39.— plane, of dihedral, 39.— of skew lines, 113.— measure of, 38, 87.— of two planes, cosine, 70.— parallel, 106, 107.

Angles of a triangle, 31.

— exterior of a triangle, 31.— Clifford, 126.

Archimedes, 24.

Area, 170, 175, 178, 211.— of a circle, 178.

— of a plane, 178.

— of a polygon, 178.— of a triangle, 175, 176, 177.

Aionhold, 159.

Asymptotes, 152.

Asymptotic lines, 196, 202, 203,

212 213Author, 116, 127, 130, 154, 156,

158, 167, 226, 230, 232, 234.

Axes, co-ordinate, 64, 67.

Axial plane of sphere, 138.

Axis of a circle, 131, 134, 135, 150.

— radical of two circles, 134, 135,

136.— of a conic, 143.

Axis of a chain, 119.— of a pencil of complexes, 116.

Barbarin, 154.

Battaglini, 131.Beck, 116.

Beltrami, 67, 210.Bianchi, 6, 187, 188, 204, 206, 210,

226, 280, 281.Birectangnlar quadrilateral, 43, 44,

Bisector of an angle, 102, 103, 109,138, 135, 186, 143, 146, 153, 157159, 220, 222.

Bolza, 209.

Borel, 34.

Bound of half-line, 28.

Bound of half-plane, 30.

Bromwich, 154.

Canal surface, 156.

Cayley, 88, 97, 157.

Central conic, 143-153.Central qnadric, 157-60.Centre of a circle, 135, 136, 137.— of a conic, 143, 148, 149, 150.— of gravity of points, 102, 108,

109, 138, 135, 136, 143, 146, 153,159, 220, 222.

Centre of quadric, 157.— of similitude, 134, 135, 136.

Ceva, 105.

Chain congruence, 121, 129.— of crosses, 119, 120, 128.

Circle, 131-137, 143, 151, 178, 188.— auxiliary to conic, 152.

Clebsch, 159, 176.

Clifford, 99, 126, 129, 156, 157, 205,212, 240.

Coaxal pencil of complexes, 116,

124.

Coaxality, 20.

Collinearity, 18, 102, 103, 104, 105,

134, 136, 251.

288 INDEX

CoUineations, 29, 38, 69, 70, 94, 119,

127, 239, 265, 266, 268.

Comparableness of angles, 34, 35.

Complex of lines, 116.

Concurrence, 18, 102, 103, 105, 134,

186, 251.

Cone of revolution, 185.

Confocal conies, 153.

Confocal quadrics, 160, 164.

Conformal transformations, 198.

Congruence of distances, 14, 15, 16,

17, 28, 86, 79.— of segments, 28.— of angles, 31, 33, 34, 36, 38, 39.

- of triangles, 31, 32.

— synectic, 120, 122.

— chain, 121, 129.— of lines, analytic, 215-235.— of lines, general, 218.— of normals, 162, 208, 210, 222,

223, 224, 225, 226, 227, 229, 235.— of normals, to surfaces of Graus-

sian curvature zero, 123, 208, 226,

227 235— isotropic, 164, 226, 227, 230, 232,

234, 235.

Congruent figures, 28.

Congruent transformations, 29, 87,

38, 69, 70, 73, 74, 80, 82, 92-100,

239, 268, 269, 270, 271, 278, 279,

280.

Conic, 142-53, 272.

Conic, eleven-point or line, 147.

Conjugate diameters ofa conic, 148.

— directions on a surface, 195,196.— harmonic, 252, 253, 254, 257,

259, 261.

Connectivity of space, 238.

Consistent region, 78, 79, 80, 83,

236, 237, 238.

Continuity, axiom of, 23, 24, 75, 249.— in change of angles and sides ofa triangle, 40, 41, 42.

CoH>rdinates of a line, 110, 264.— of a point, 64, 68, 176, 187, 188,

194, 236, 237, 263, 264, 275.— of a plane, 264.

Coplanarity, 109, 138.

Cosine of angle, 54, 70, 279.— of distance, 52, 285.

Cosines, direction, 67, 69, 278.— law of, 57.

Cross, 117, 118, 119, 124, 125, 281.Cross ratios, 73, 86, 88, 89, 90, 91,

247, 259, 260, 261, 262, 264, 265.

Cross space, 118.

Curvature of a curve. 133, 188, 189,

200, 201.— Gaussian, 67, 123, 130, 204, 205.

206, 207, 208, 275, 281, 282, 283.

— geodesic, 208, 209.— mean relative, 200, 212.— total relative, 200, 203. 204, 205.— of space, 53, 176, 189, 204, 275,

281.— lines of, 198, 199.— surfaces of zero, 123, 204, 205,

206, 207, 208, 226, 227, 235,

Dannmeyer, 170.

Darboux, 141. 212, 278.

Dehn, 46, 181.

Density of segment, 16.

Desargues, 75, 146, 251.

Desmic configuration, 108, 109, 110.

138.

Diagonal points of quadrangle, 252.

Diagonals of quadrilateral, 252.

Diameters of conic, 148, 149, 150,

151.

— of quadric, 159, 160.

Difference of distances, 17, 85.

Director points and directrices, 144,

145, 146.

Discrepancy of a triangle, 46, 174.

Distance, 13, 72, 73, 74, 76, 78, 87,

89, 90, 91, 272, 273, 285.

Distance, directed, 62, 66, 90.

Distance of two points, cosine, '52.

69, 78, 285.— from point to plane, 70.

— of skew lines. 111, 112, 114.— element, 66, 67, 187, 194, 276-84.

Division of segment, 24, 2.5. 26, 27.

Dunkel, 60.

Dupin, 141, 197, 201, 205.

Edge of tetrahedron, 20.

Ellipse, 142, 148, 146, 158, 167, 168.

169.

Ellipsoid, 154. 156, 167, 168, 169.

Elliptic co-ordinates, 158, 161.— hypothesis, 46, 73, 74, 274, 285.— space, 82, 83, 245.

Engel, 43.

Enlargement of congruent trans-

formation, 29.

Enriqnes, 33, 177, 247.

Equidistant curves, 182, 148.— surfaces, 156.

INDEX 289

Equivalent points, 81, 82, 236.Euclid, 47, 72.

Euclidean hypothesis, 46, 72, 73,274 285

— spsice, 77, 91, 239, 240, 241, 242.Evolutes, 192, 193, 194.

Excess of a triangle, 174, 175, 176,

177.

Extension of segment, 15, 79.— of space, 77, 78, 79, 80.

Extremity of segment, 15.

Face of tetrahedron, 20.

Fibbi, 215, 221.

Focal cones, 158, 159.— conies, 158, 159, 167, 168, 169.— Unes, 144, 145, 146, 147, 151.— points and planes, 221, 222, 223,

224.— surfaces, 210, 226, 232.

Foci, 144, 145, 146, 147, 151.

Forms, fundamental one-dimen-sional, 259, 260, 261, 267.

Von Frank, 186.

Frenet, 190.

Fricke, 244.

Frischauf, 176, 186.

Fubini, 227, 229.

Fundamental region, 239-46.— one-dimensional forms, 259, 260,

261, 267.

Geodesic curvature, 208, 209.

— lines, 163, 209, 210, 274-81, 284,

285— surfaces, 279, 280, 281.

Gerard, 48, 53.

Graves, 153.

Greater than, 15, 16, 17, 34, 35,

37, 92.

Half-line, 28-33, 38, 64, 67.

Half-plane, 30, 37, 38, 39.

Halsted, 24, 75, 177.

Hamilton, 98, 120, 221.

Harmonic conjugate, 252, 253, 255,

261.— separation, 252, 253, 254, 257.

— set, 252, 253.

Hilbert, 13, 24, 36, 75, 177, 265.

Homothetic conies, 152, 153.

— quadrics, 160.

Horocycle, 132, 143, 243.

Horocyclic surface, 156, 205.

Hyperbola, 142, 146, 167, 168, 169.

COOZ.1DOE

Hyperbolic hypothesis, 46, 72, 73,

78, 274, 285.— space, 78, 236.

Hyperboloid, 155, 167, 168, 169.

Ideal elements, 84, 85.

Imaginary elements, 86, 87, 266,267.

Improper cross, 117, 118, 127, 231.— ray, 231, 23?.

Indicatrix of Dupin, 201, 205.Infinitely distant elements, 84, 85-

Infinitesimal domain, 42, 47, 68,

174, 175.

Initial point, 62.

Intersection of lines, 17, 249.— of planes, 22, 251.

Involution, 86, 87, 266, 267.

Isosceles quadrilateral, 43, 50.— triangle, 32, 34.

Isotropic curves, 203, 209.— congruence, 164, 226, 227, 230,

232, 234, 235.

Joachimsthal, 197.

Jordan, 277.

measure of curvature of space.

53, 176, 189, 204, 275, 281.

Killing, 142, 237, 245.

Klein, 97, 129, 161, 244.

Kummer, 215.

Layer of cross space, 118, 119, 125.

Left and right generators of Abso-lute, 99, 124, 125, 234.

Left and right translations, 99, 100,

245.

Left and rightparataxy, 99, 208, 225.

Length of arc, 276.

Less than, 15, 16, 17, 34, 35, 37, 92.

Levy, 13, 75.

Lie, 268, 270, 271.

Liebmann, 142.

Limiting points and planes, 219220, 222.

Lindemann, 139, 176.

Line, 17, 78, 248, 249.

Lobatchewsky, 46, 106.

Lobatchewskian hypothesis, 46.

Luroth, 87, 89, 267.

Manning, 107, 176, 205.

Marie, Ste-, 47.

290 INDEX

Measure of distance, 27, 28, 87.

— of curvature of space, 33, 176,

189, 204, 207, 275, 281.

Menelaus, 105.

Meunier, 201, 208.

Middle point of segment, 24.

Minimal surfaces, 129, 210-14.Moment, relative of two lines, 112.

— relative of two rays, 115, 192.

Moore, 13, 46, 75. .

Motions, 97, 98, 99.

Multiply connected space, 238-46Mfinich, 139.

Normals to curve, 192, 193, 194.

— to surface, 162, 197, 208, 210, 222,

223, 224, 225, 226, 227, 229, 235.

Null angle, 30.

— distance, 14.

Opposite edges of tettahedon, 20.— half-lines, 31.

— senses, 63, 86, 266.-

— sides of plane, 22.

Origin, 64.

Orthogonal points, 101, 103, 118,

132, 135, 136, 137, 138, 139, 143,

189, 205, 215, 217, 219, 224.

Orthogonal substitutions, 69, 70, 73,

97, 98.— system of surfaces, 197, 198.

d'Ovidio, 112, 142, 170.

Padoa, 13, 254.

Parabola, 142, 143.

Parabolic hypothesis, 46.

Paraboloid, 155, 157.

Parallel angle, 106, 107, 110.

ParalleUsm, 85, 99, 106, 113, 234,235

Parataxy, 99, 114, 125, 129, 206,207, 208, 225, 233, 234, 235.

Pasch, 13, 29, 86, 265.

Peano, 13.

Pencil of complexes, 116.— of geodesies, 279.

Perpendicularity, 34, 36, 37, 39,

101, 103, 118, 132, 135, 136, 137,

138, 139, 143, 182, 183, 193, 197,

217, 219, 220, 224, 279.

Phi function, 50, 51, 52.

Picard, 175, 210.

Fieri, 13, 74, 86, 247.

Plane, 20, 21, 22, 38, 67, 70, 81, 82,

95, 109, 110, 118, 224, 242, 243,

249, 250, 251, 253, 259, 264, 265,

268, 269.

Poincar^, 139.

Point, 13, 78, 84, 86, 247, 266,

275.

Polyg;on, 178.

Principal points and planes, 220.

Products connected with a conic,

145, 149, 150.

Projection, 253, 260.

Projectivity, 259, 260, 262, 267.

Pseudo-isotropic congruence, 229,

230, 234, 235.

Pseudo-normal congruence, 224,

229.

Pseudo-parallelism of lines, 113,

234, 235.

Pythagorean theorem, 55, 57.

Quadrangle, complete, 252.

Quadrilateral, 43, 44, 49, 174.— complete, 252, 253, 266.

Quaternions, 98, 245.

Ratio of opposite sides of quadri-

lateral, 49, 50, 51, 52, 53.

Ratios, constant connected withconies, 144, 151.

Ray, 114, 115, 191, 192, 227, 228,

234, 235.

Rectangle, 43, 44, 45, 46.

Reflection in plane, 39, 82.— in point, 62.

Region consistent, 78, 79, 80, 81,

83, 236, 237, 238.

Region, fundamental, 239-46.— restricted, 269, 276, 277, 278.

Revolution, surfaces of, 155, 156.

Riccordi, 131.

Richmond, 184.

Riemann, 46, 53, 67, 275.

Riemannian hypothesis, 46.

Right angle, 32, 34, 39, 279.— triangle, 32, 44, 45, 55.

Right and left generators of Abso-lute, 94, 124, 125, 234.

Right and left parataxy, 99, 208,ado.

Right and left translations, 99, 100,245.

Russell, 74.

Saccheri, 43, 50.

Salmon, 134.

Scalene triangles, 34, 35.

INDEX 291

Schlafli, 183, 184, 185.Schur, 13, 275.

Segment, 15, 16, 17, 18, 23, 24, 25,26, 28.

Segre, 119.

Semi-hyperbola, 142.

Semi-hyperboloid, 154.

Sense of directed distances, 63, 64.— of description of involution, 86,

266.

Separation, 248, 249, 255, 256, 257,

258, 259, 262.— classes, 247, 249, 255, 257, 258.— harmonic, 252, 253, 254, 257.

Sides of angle, 30, 31.— of quadrangle, 252.— of quadrilateral, 43, 252.— of triangle, 19, 31, 32, 35, 36.

Similitude, centres of, 134, 135,

136.

Sine of distance from point to plane,70.

Sines, law of, 58, 59.

Singular region, 238.

Space, 20, 21, 22, 78, 238-46, 250,251.

Sphere, 73, 74, 138-41, 156, 227.

228.

Spheres, representing, 227, 228.

Spherical space, 83.

Spheroid, 155, 156.

Stackel, 43.

Staude, 162.

Von Staudt, 86, 87, 89, 267.

Stephanos, 108.

Stolz, 24.

Story, 142.

Strip, 128, 129.

Study, 91, 93, 99, 116, 123, 125, 126,

229, 234. *

Sturm, 233.

Sum of angles, 32, 34.

— of angles of a triangle, 45, 46.

Sum of distances, 14-17, 92, 93.

Sum of distances connected with a

conic, 145, 148, 149.

Sum of distances connected with a

quadric, 160.

Sum of two sides of triangle, 35.

Supplementary angles, 32.

Surface integral, 175.

Symmetry transformations, 98, 99,127.

Synectic congruence, 120, 122.

Tait, 286.

Tangent plane to surface, 194, 195.Tannery, 273.

Terminal point, 62.

Tetrahedron, 20, 21, 181, 182, 183.Tensor, 98.

Thread construction, 169.Torsion, 190, 191, 192, 203, 207.Transformations, congruent, 29, 37,

88, 69, 70, 73, 74. 80, 82, 92-100,239, 268, 269, 270, 271, 278, 279,280.

Translations, 62, 63, 100, 128, 239,240, 245, 246.

Triangle, 18, 19, 31-5, 170, 172,

174, 175, 176, 177.

Triangles, congruent, 31.

Trirectangular quadrilateral, 48.

Ultra-infinite elements, 85, 187.

Umbilical points, 162.

Vahlen, 13, 24, 75, 247, 260, 265.Vailati, 248, 254.

Veblen, 13, 19, 76, 247.Veronese, 13, 74.

Vertex of angle, 30, 31.— of quadrangle, 252.— of quadrilateral, 252.— of tetrahedron, 20.— of triangle, 19.

Vertical angles, 32, 84.

Volume, 181, 182.— integral, 182.— of cone, 185.— of sphere, 186.— of tetrahedron, 182, 188, 184,

185.

VoBs, 188.

Weber, 46.

Weierstrass, 142.

Within a segment, 15, 18.

Within a triangle, 19.

Woods, 237, 245, 246, 279.

Young, 247.

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